On the integrated density of states of random Pauli Hamiltonians

Naomasa Ueki

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan (e-mail: ueki@math.h.kyoto-u.ac.jp)

ABSTRACT

The difference of the integrated densities of states (IDS) of the two components of a random Pauli Hamiltonian is shown to equal a constant given in terms of the expectation of the magnetic field. This formula is a random version of the Aharonov and Casher theory or that of the Atiyah and Singer index theorem. By this formula, the IDS is shown to jump at 0 if the expectation of the magnetic field is nonzero. For simple cases where the expectation of the magnetic field is zero, a lower estimate of the asymptotics of the IDS at 0 is given. This lower estimate shows that the IDS decays slower than known results for random Schrödinger operators whose infimum of the spectrum is 0. Moreover the strong-magnetic-field limit of the IDS is identified in a general setting.

1. Introduction

Let B = (B_{ω}(x)) (ω ∈ Ω, x ∈ ℝ^{2}) be a real random field such that (a-i)B_{ω}(x) is stationary and ergodic with respect to the shift in the variable x; (a-ii) the sample path ℝ^{2} ∋ x ↦ B_{ω}(x) is continuous; (a-iii) there exists α > 0 such that E[exp(α|B_{ω}(0)|)] < ∞; (a-iv) there exists an ℝ^{2}-valued random field A = (A_{ω}^{j}(x))_{j=1}^{2} such that ℝ^{2} ∋ x ↦ A_{ω}(x)

is continuous, ∂_{1}A_{ω}^{2}(x) − ∂_{2}A_{ω}^{1}(x) = B_{ω}(x) and ∂_{1}A_{ω}^{1}(x) + ∂_{2}A_{ω}^{2}(x) = 0 in the sense of distributions.

For each ω, regarding x ↦ B_{ω}(x) as a magnetic field, and x ↦ A_{ω}(x) as its vector potential, we consider a Pauli Hamiltonian H_{ω} = H_{ω}^{+} ⊕ H_{ω}^{−} formally defined by

H_{ω}^{±} = ∑_{j=1}^{2}(i∂_{j} + A_{ω}^{j}(x))^{2} ± B_{ω}(x),  (1.1)

where i = \sqrt{−1} and ∂_{j} = ∂/(∂x^{j}). Let N^{±}(λ), λ ∈ ℝ, be the integrated density of states of the operators H_{ω}^{±}. We define those as right continuous functions. For the exact definition, see Section 2 below.

The first result in this paper is the following:

Theorem 1. Under the conditions (a-i)–(a-iv), we have

N^{−}(λ) − N^{+}(λ) = E[B_{ω}(0)]/(2π)  (1.2)

for any λ ≥ 0.

This is an extension of the index theorem to a noncompact setting in terms of the integrated density of states. As in the usual index theorem, a main tool for the proof is the theory on the supersymmetry (cf. Section 6.3 in [18]): by the supersymmetry we have

N^{−}(λ) − N^{+}(λ) = Ñ^{−}(t) − Ñ^{+}(t)  (1.3)

for any λ ≥ 0 and t > 0, where Ñ^{±}(t) is the Laplace-Stieltjes transform of N^{±}(λ). Then (1.2) is obtained by identifying the limit of the right hand side of (1.3) represented in terms of the heat semigroup as t ↓ 0. This is the same technique for the proof of the index theorem (cf. [4], [18], [33], [43]). However in our case, we should prove that the boundary condition to define the integrated density of states does not affect the result. We carry out this in a general setting and this is our main contribution. For this, see the proof of Lemma 4.1 and Remark 4.1 below.

From this theorem, we easily know on the kernels as follows:

Corollary 1. Under the conditions (a-i)–(a-iv), we have

N^{±}(0) ≥ E[B_{ω}(0)]_{∓}/(2π),  (1.4)

where a_{±} = max{±a, 0} for any a ∈ ℝ. Therefore, if ∓E[B_{ω}(0)] > 0, then 0 is the eigenvalue of H_{ω}^{±} with infinite multiplicity.

Corollary 2. Under the conditions (a-i)–(a-iv) and H_{ω}^{+} ≥ b for some b ∈ (0, ∞), we have

N^{−}(0) = E[B_{ω}(0)]/(2π)  (1.5)

and

inf{λ : N^{+}(λ) ≠ 0} = inf{λ : N^{−}(λ) − N^{−}(0) ≠ 0} ≥ b.  (1.6)

Example 1.1. (i) If b_{1} := ess sup_{ω∈Ω} inf_{x∈ℝ^{2}} B_{ω}(x) > 0, then we have (1.5) and (1.6) with b = 2b_{1} by the well known estimate H_{ω}^{+} ≥ 2B_{ω}(x) (cf. Avron, Herbst and Simon [7] Theorem 2.9).

(ii) If there exist a random variable B_{ω}^{0} and a random field φ_{ω}(x) such that φ_{ω}(x) is C^{2} in x ∈ ℝ^{2}, B_{ω}(x) = B_{ω}^{0} + Δφ_{ω}(x) and b_{2} := ess sup_{ω∈Ω}B_{ω}^{0} exp(−2osc φ_{ω}) > 0, then we have (1.5) and (1.6) with b = 2b_{2} by a Raikov’s result (Proposition 1.2 in [50]), where Δ is the Laplacian and osc φ_{ω} = sup_{x∈ℝ^{2}} φ_{ω}(x) − inf_{x∈ℝ^{2}} φ_{ω}(x). If the first derivatives of φ_{ω}(x) are almost surely bounded on ℝ^{2}, then B_{ω}^{0} is not random and coincides with the expectation E[B_{ω}(0)] by the ergodic theorem and Green’s formula.

These results are extended to arbitrary even dimensional space as Theorem 4 below. In this paper we prove the general theorem.

On the dimension of the kernel of the deterministic Pauli Hamiltonian H_{0} = H_{0}^{+} ⊕ H_{0}^{−} obtained by replacing (A_{ω}^{j}(x))_{j=1}^{2} and B_{ω}(x) in (1.1) by a deterministic vector potential (A_{0}^{j}(x))_{j=1}^{2} and its magnetic field B_{0}(x), respectively, we have Aharonov and Casher’s theory [2]. Their theory gave a basis constituting

⌊\frac{1}{2π} ∫_{ℝ^{2}} B_{0}(x)dx⌋

elements of the kernel of H_{0}^{−} under the compactness of suppB_{0} and

1 ≤ \frac{1}{2π} ∫ B_{0}(x)dx < ∞,

where ⌊a⌋ is the largest integer smaller than a for any a > 0 (cf.[10], [18]). Their theory has been extended to many situations where B_{0}(x) is integrable in x. For this aspect, see [8], [10], [18], [24], [25] and the references therein. For the case that B_{0}(x) is periodic as B_{0}(x^{1} + T^{1}, x^{2}) = B_{0}(x^{1}, x^{2} + T^{2}) = B_{0}(x), Dubrovin and Novikov [21], [22] gave a Bloch basis and showed the existence of the gap between zero and the rest of the spectrum under the condition that

\frac{1}{2π} ∫_{0}^{T^{1}} ∫_{0}^{T^{2}} B_{0}(x)dx^{1}dx^{2}  (1.7)

is a natural number. From their result, we see that N_{0}^{−}(0) equals the natural number in (1.7) and inf{λ : N_{0}^{−}(λ) ≠ N_{0}^{−}(0)} > 0, where N_{0}^{−}(λ) is the integrated density of states of H_{0}^{−} (cf. [29]). For the case that

B_{0}(x) = B_{0} + B̃_{0}(x)  (1.8)

is almost periodic such that B_{0} > 0,

B̃_{0}(x) = Re ∑_{n=1}^{∞} B_{n}e^{iC_{n}x},

and

∑_{n=1}^{∞} |B_{n}|(|C_{n}|^{−2} + 1) < ∞,

Raikov [50] showed the same formulas as (1.5) and (1.6) where N^{±}, E[B_{ω}(0)] and b are replaced by N_{0}^{±}, B_{0} and 2B_{0} exp(−2osc φ), respectively, where

φ(x) = Δ^{−1}B̃_{0}(x) = Re ∑_{n=1}^{∞} \frac{−B_{n}}{C_{n}^{2}} e^{iC_{n}x}.

Since

lim_{R→∞} \frac{1}{R^{2}} ⌊|\frac{1}{2π} ∫_{Λ(R)} B_{ω}(x)dx|⌋ = \frac{E[B_{ω}(0)]}{2π}

for an ergodic random field B_{ω}(x) and

lim_{R→∞} \frac{1}{R^{2}} ∫_{Λ(R)} B_{0}(x)dx = B_{0}

for a function B_{0}(x) given by (1.8), our result is regarded as an extension of [2], [21], [22] and [50] to a general stationary ergodic magnetic field.

We next consider a strong-magnetic-field asymptotics: we consider the asymptotics as ξ → ∞ of the integrated density of states N^{±}(λ; ξ) of the operator formally defined by

H_{ω}^{ξ,±} = ∑_{j=1}^{2}(i∂_{j} + ξA_{ω}^{j}(x))^{2} ± ξB_{ω}(x).  (1.9)

We assume (a-v) there are c > 0 and ν > 1 such that E[exp(α|B_{ω}(0)|)] ≤ c exp(α^{ν}) for any α > 0. Then by the localizing effect of the strong magnetic field, we can determine the leading

term:

Theorem 2. Under the conditions (a-i), (a-ii), (a-iv) and (a-v), the integrated density of states N^{±}(λ; ξ) of the operators H_{ω}^{ξ,±} satisfies

lim_{ξ→∞} \frac{N^{±}(λ; ξ)}{ξ} = \frac{E[B_{ω}(0)_{∓}]}{2π}  (1.10)

for each λ > 0.

In the right hand side of (1.10), the positive or negative part is taken inside the expectation whereas that is taken outside of the expectation in (1.4). This is due to the localizing effect of the strong magnetic field. The right hand side of (1.10) may be greater than that of (1.4). However we cannot estimate the value of N^{±}(0) by (1.10), since the equation (1.10) is not proven uniformly on an interval including 0.

The strong-magnetic-field asymptotics of the integrated density of states has been extensively studied for the Pauli Hamiltonian with the deterministic constant magnetic field perturbed by a random electric scalar potential [11], [38], [39], [41], [49], [54], [55], [56]. For variable magnetic fields, we have related deterministic results [23], [48]. In [48], Raikov gives the leading term of the counting function of the number of the eigenvalues. Moreover in [38], [48] and [49], the three dimensional cases are also treated, where the degeneracy of the magnetic field brings another problem.

In this paper we concentrate only on the unperturbed Pauli Hamiltonian with a nondegenerate magnetic field and prove the fundamental result in a considerably general situation: we prove a generalization of Theorem 2 to arbitrary even dimensional space. The result is Theorem 5 below.

To prove the upper estimate, we use a method of Erdös [23] proving

lim_{ξ→∞} \frac{1}{ξ}E(\{0\}; 0, 0; ξ) = \frac{|B_{0}(0)|}{2π}  (1.11)

under some conditions, where E(Λ; x, y; ξ), Λ ∈ B(ℝ), x, y ∈ ℝ^{2}, ξ > 0, is the integral kernel of the resolution of the identity of the deterministic Pauli Hamiltonian defined by H_{0}^{ξ} = H_{0}^{ξ,+} ⊕H_{0}^{ξ,−} obtained by replacing (A_{ω}^{j}(x))_{j=1}^{2} and B_{ω}(x) in (1.9) by a deterministic vector potential (A_{0}^{j}(x))_{j=1}^{2} and its magnetic field B_{0}(x), respectively. By the same proof, we can also show

lim_{ξ→∞} \frac{1}{ξ}E^{±}([0, λ]; 0, 0; ξ) = \frac{B_{0}(0)_{∓}}{2π}

for each λ > 0, where E_{ω}^{±}(Λ; x, y; ξ) is the integral kernel of the resolution of the identity of H_{0}^{ξ,±}. This result relates with (1.10) by the representation N^{±}(λ; ξ) = E[E_{ω}^{±}([0, λ]; 0, 0; ξ)], where E_{ω}^{±}(Λ; x, y; ξ) is the integral kernel of the resolution of the identity of the random Pauli Hamiltonian H_{ω}^{ξ,±} (cf. [13]). To show the upper estimate for (1.11), he determined the limit of a function defined by using the corresponding integral kernel of the heat semigroup represented by the Feynman-Kac-Itô formula. Since the Laplace-Stieltjes transform of N^{±}(λ; ξ) also has a similar representation (see Lemma 3.1 below), we can show the upper estimate by the same method.

To prove the lower estimate, we estimate the number of low lying eigenvalues of the Hamiltonian restricted to small domains. For this we estimate the energy of the functions obtained by restricting the functions in Aharonov and Casher’s theory to small domains. For the higher dimensional case, we need extra work to show the lower estimate: to construct the functions in Aharonov and Casher’s theory, we use the theory of the \overline{∂}- Neumann problem [9]. For this we assume that the magnetic field is derived from a skew-Hermitian matrix valued random field. Moreover our estimate of the number of low lying eigenvalues of the restricted Hamiltonian becomes best when the domain is a polydisc depending on ω. Then, to obtain a sharp estimate of N(λ; ξ), we should take the osculatory packing so that the related quantities are measurable.

We next consider the low energy asymptotics of N^{±}(λ). For this, Casher and Neuberger [15] showed

N^{±}(λ) ∼ c\sqrt{λ} as λ ↓ 0,  (1.12)

when B_{ω}(x) is a Gaussian white noise with mean zero, by a heuristic argument using Aharonov and Casher’s theory and a techniques in the field theory. The equation (1.12) means that (1.5) may hold without the strict positivity of H_{ω}^{+} and that (1.10) may not hold at λ = 0. More interesting point of (1.12) is that the decay of N^{±}(λ) as λ ↓ 0 is slower than that of the free Hamiltonian −Δ. This behavior is in contrary to those of other random Schrödinger operators (cf.[14], [46]). The same type of behavior was also shown by Comtet, Georges and Le Doussal [16], [17], where

N^{±}(λ) ∼ c(log(1/λ))^{−3} as λ ↓ 0.  (1.13)

is shown when A_{ω}^{1}(x) is a Gaussian white noise depending only on x^{1} and A_{ω}^{2} = 0 by a heuristic argument reducing the problem to a 1-dimensional problem and representing N^{±}(λ) in terms of Bessel functions. More recently, Ludwig, Fisher, Shankar and Grinstein [40], Motrunich, Damle and Huse [44] and Fukui [27] investigated the asymptotics of the density of states of the Dirac operator by various heuristic arguments when A_{ω}^{1} and A_{ω}^{2} are independent Gaussian white noises. In this case, their theory implies

N^{±}(λ) ∼ cλ^{α} as λ ↓ 0,  (1.14)

where α is a constant in the interval (0, 1) determined by the covariance of the white noise. The same behavior was also shown by a heuristic argument in Horovitz and Le Doussal [30] when A_{ω}(x) = (−∂_{2}C_{ω}(x), ∂_{1}C_{ω}(x)) and C_{ω}(x) is a Gaussian random field such that E[(C_{ω}(x) − C_{ω}(x′))^{2}] ∼ log |x − x′| as |x − x′| → ∞.

In this paper, we give a rigorous lower estimate proven by slightly modifying the argument used in the lower estimate for (1.10). We consider simple cases as follows:

(a-vi) B_{ω}(x) = B_{ω}^{1}(x^{1}) + B_{ω}^{2}(x^{2}), where B_{ω}^{1}(x^{1}) and B_{ω}^{2}(x^{2}) are independent Gaussian random processes with the mean zero and the covariance β^{j}(x^{j}) = E[B_{ω}^{j}(x^{j})B_{ω}^{j}(0)], j = 1, 2 satisfying 0 ≤ β^{j}(x^{j}) ∈ L^{1}(ℝ) and inf_{|x^{j}|≤r} β^{j}(x^{j}) > 0 for some r > 0.

(a-vii) B_{ω}(x) = B_{ω}^{1}(x^{1}), where B_{ω}^{1}(x^{1}) is a Gaussian random process with the mean zero and the covariance β^{1}(x^{1}) = E[B_{ω}^{1}(x^{1})B_{ω}^{1}(0)] satisfying 0 ≤ β^{1}(x^{1}) ∈ L^{1}(ℝ) and inf_{|x^{1}|≤r} β^{1}(x^{1}) > 0 for some r > 0.

In these cases, we have the following:

Theorem 3. Under the conditions (a-i), (a-ii) and either (a-vi) or (a-vii), the integrated density of states N^{±}(λ) of the operators H_{ω}^{±} satisfy

lim inf_{λ↓0} N^{±}(λ)(log(1/λ))^{1/3} > 0.  (1.15)

This theorem indicates that our integrated density of states decays slower than those in any known results for random Schrödinger operators whose infimum of the spectrum is 0. We prove this theorem in Section 6 below.

The organization of this paper is as follows. In the next section, we formulate our problem in a general setting and state the general theorems, Theorem 4 and Theorem 5. In Section 3, we first recall a representation of the integrated density of states by the Feynman-Kac-Itô formula in [53] to prove an upper estimate for Theorem 5. The same representation is used to prove Theorem 4 by using also a supersymmetry in Section 4. After that, in Section 5, we prove a lower estimate for Theorem 5 by referring Aharonov and Casher’s theory. The same techniques are used in Section 6 to prove Theorem 3.

2. A general setting

Let d = 2h be a positive even number. Let B = (B_{ω,k}^{j}(x))_{1≤j,k≤d} (ω ∈ Ω, x ∈ ℝ^{d}) be a real skew-symmetric matrix valued random field such that

(b-i) B is stationary and ergodic with respect to the shift in the variable x; (b-ii) the sample path x ↦ B_{ω}(x) is continuous; (b-iii) the corresponding form is closed: d{∑_{j<k} B_{ω,k}^{j}(x)dx^{j} ∧ dx^{k}} = 0 in the sense of

distributions; (b-iv) there exists α > 0 such that E[exp(α|B_{ω,k}^{j}(0)|)] < ∞ for any 1 ≤ j < k ≤ d; We give a vector potential A_{ω} = (A_{ω}^{j}(x))_{1≤j≤d} by the Poincaré gauge:

A_{ω}^{j}(x) := ∫_{0}^{1} ∑_{k=1}^{d} B_{ω,j}^{k}(tx)tx^{k}dt.  (2.1)

This vector potential is continuous in x and satisfies

d{ ∑_{j=1}^{d} A_{ω}^{j}(x)dx^{j}} = ∑_{j<k} B_{ω,k}^{j}(x)dx^{j} ∧ dx^{k}

in the sense of distributions. Let γ_{1}, γ_{2}, …, γ_{d} be Hermitian matrices acting a 2^{h}- dimensional complex Hilbert space V and satisfying the commutation relation

γ_{j}γ_{k} + γ_{k}γ_{j} = \{ \begin{matrix}2I if j = k, \\ O if j ≠ k, \end{matrix}  (2.2)

where I and O are the identity and the zero matrices, respectively (see e.g. [18] §12.2). Then we can define the random Dirac operator by

D_{ω} := ∑_{j=1}^{d} γ_{j}(i∂_{j} + A_{ω}^{j}(x))

acting on the space C_{0}^{∞}(ℝ^{d} → V) of V -valued smooth functions with compact supports. It is known that this operator is essentially self-adjoint in the space L^{2}(ℝ^{d} → V) of V - valued L^{2}-functions (cf. [35], [34] §9.2.1). We denote the unique self-adjoint extension by the same symbol. The random Pauli Hamiltonian is the self-adjoint operator defined by H_{ω} = D_{ω}^{2}.

We use an Hermitian matrix Γ = i^{h}γ_{1}γ_{2} ⋯ γ_{d}, which depends on the orientation of ℝ^{d}. The eigenvalues of this matrix are 1 and −1. The projection to the eigenspace V_{±} = {Φ ∈ V : ΓΦ = ±Φ} is Γ_{±} = (I ± Γ)/2. Since D_{ω}Γ = −ΓD_{ω}, the operator H_{ω} is regarded as the direct sum of the two self-adjoint operators H_{ω}^{+} and H_{ω}^{−} on the space L^{2}(ℝ^{d} → V_{+}) and L^{2}(ℝ^{d} → V_{−}), respectively.

We define the integrated density of states N^{±}(λ), λ ∈ ℝ, of the random Pauli Hamiltonian H_{ω}^{±} by the usual method: for each R > 0, let Λ(R) := (−R/2, R/2)^{d}. Let H_{ω}^{±,R} be the self-adjoint operator on the space L^{2}(Λ(R) → V_{±}) with the Dirichlet boundary condition: H_{ω}^{±,R} is the self-adjoint operator corresponding to the closure of the quadratic form q_{ω}(Φ, Ψ) := ⟨D_{ω}Φ(x), D_{ω}Ψ(x)⟩_{V} with the domain C_{0}^{∞}(Λ(R) → V_{±}), where ⟨·, ·⟩_{V} is the Hermitian inner product of the space L^{2}(ℝ^{d} → V). Then the spectra of H_{ω}^{±,R} are purely discrete. Let N_{ω}^{±,R}(λ) be its counting function, which is the number of eigenvalues of H_{ω}^{±,R}(λ) not exceeding λ. Then, by the method by Kirsch-Martinelli (cf. [14], [37]), we can show the existence of a deterministic, right continuous, increasing function N^{±}(λ) such that

R^{−d}N_{ω}^{±,R}(λ) ⟶ N^{±}(λ)  (2.3)

as R → ∞ for any point of continuity of N^{±}(λ) and almost all ω. We here remark that the method by Kirsch-Martinelli cannot be applied directly since the vector potential A_{ω}(x) appearing in the Hamiltonian is not stationary. However this difficulty can be overcome by using the gauge invariance as in Lemma 3.1 of [53].

Remark 2.1. (i) As is stated in Remark 2.1 in [53], we can use other boundary conditions to define the integrated density of states: if we define an operator Ĥ_{ω}^{±,R} with a Neumann boundary condition by the corresponding self-adjoint operator to the quadratic form

q̂_{ω}(Φ, Ψ) := ∑_{j=1}^{d}⟨(i∂_{j} + A_{ω}^{j}(x))Φ(x), (i∂_{j} + A_{ω}^{j}(x))Ψ(x)⟩_{V}

+⟨Φ(x), ∑_{j<k} iγ_{j}γ_{k}B_{ω,k}^{j}(x)Ψ(x)⟩_{V}  (2.4)

with the domain

W^{1,2}(Λ(R) → V_{±})
:= {Φ ∈ L^{2}(Λ(R) → V_{±}) : ∂_{j}Φ ∈ L^{2}(Λ(R) → V_{±}) for j = 1, 2, …, d}

and denote the corresponding counting function of the spectra by N̂_{ω}^{±,R}(λ), then we have

R^{−d}N̂_{ω}^{±,R}(λ) ⟶ N^{±}(λ)  (2.5)

as R → ∞ for any point of continuity of N^{±}(λ) and almost all ω. For the details, refer [19], [31], [32] and [45].

(ii) To define the integrated density of states by a Neumann boundary condition, it is important to use the form q̂_{ω} instead of q_{ω}, since the spectrum of the self-adjoint operator H̃_{ω}^{±,R} corresponding to the quadratic form q_{ω} with the domain

Q := {Φ ∈ L^{2}(Λ(R) → V_{±}) : ∑_{j=1}^{d} γ_{j}∂_{j}Φ ∈ L^{2}(Λ(R) → V_{±})}

may not be discrete. In fact, when d = 2 and the magnetic field is a positive constant B, the Dirac operator and the above domain are represented as

D = −2i ( \begin{matrix}0 & ∂/(∂z) − B\overline{z}/4 \\ ∂/(∂\overline{z}) − Bz/4 & 0 \end{matrix} )

and

Q = \{Φ = ^{t}(Φ_{1}, Φ_{2}) : Φ_{1}, Φ_{2}, ∂Φ_{1}/(∂\overline{z}), ∂Φ_{2}/(∂z) ∈ L^{2}(Λ(R))\},

respectively, in terms of the complex coordinate z = x^{1} + ix^{2} by the representation of {γ_{j}}_{j} in (2.12) below. Then the functions

Φ_{n} = ^{t}(z^{n} exp(−B|z|^{2}/4), 0), n ∈ ℕ,

satisfy DΦ_{n} = 0 and belong to Q. Thus they belong to the kernel of the Hamiltonian. Moreover they are linearly independent by the uniqueness theorem in the complex analysis. Therefore the dimension of the kernel of the Hamiltonian is infinity. This fact does not contradict a general theory on the discreteness of the spectrum of an elliptic differential operator on a bounded domain (cf. [1] Theorem 14.6). In fact the domain of the Hamiltonian does not satisfy the conditions in Theorem 14.6 in [1].

For any real skewsymmetric matrix B = (B_{k}^{j})_{1≤j,k≤d}, the Pffafian, Pff(B), is defined by

Pff(B) = \frac{1}{2^{h}h!} ∑_{σ∈𝔖(d)}(sgnσ)B_{σ(2)}^{σ(1)}B_{σ(4)}^{σ(3)} ⋯ B_{σ(d)}^{σ(d−1)},

where 𝔖(d) is the set of all permutation of {1, 2, …, d} and sgnσ is the signature for each permutation σ (cf. [18] Definition 12.15).

Then Theorem 1 is extended as follows:

Theorem 4. Under the conditions (b-i)–(b-iv), let N^{±}(λ) be the integrated density of states of the random Pauli Hamiltonian H_{ω}^{±}. Then it holds that

N^{+}(λ) − N^{−}(λ) = (\frac{−1}{2π})^{h}E[Pff(B_{ω}(0))]  (2.6)

for each λ ≥ 0.

We prove this theorem in Section 4.

We next state a generalization of Theorem 2. For this we introduce the following conditions:

(b-v) there exist c > 0 and ν > 1 such that E[exp(α|B_{ω,k}^{j}(0)|)] ≤ c exp(α^{ν}) for any α > 0 and 1 ≤ j < k ≤ d;

(b-vi) P (rankB_{ω}(0) = d) > 0;

(b-vii) there exist a complex structure valued random variable J_{ω} = (J_{ω,k}^{j})_{1≤j,k≤d} and R_{0} > 0 such that J_{ω}B_{ω}(x) = B_{ω}(x)J_{ω} for any x ∈ Λ(R_{0}) and the Hermitian matrix B_{ω}(0)/i is nonnegative definite on the complex vector space (ℂ^{d})_{ω}^{(1,0)} := {v ∈ ℂ^{d} : J_{ω}v = iv}.

We use the condition (b-vii) only to prove the lower estimate for the strong-magnetic-field asymptotics: in Lemma 5.1 below, we use the theory of the Dolbeault complex to solve the Dirac equation locally. This condition is equivalent with the following:

(b-vii)′ there exist a d×d orthogonal matrix valued random variable U_{ω} = (U_{ω,k}^{j})_{1≤j,k≤d} and R_{0} > 0 such that the skew-symmetric matrix valued random field B′_{ω}(x) = (B_{ω,k}^{′,j}(x))_{1≤j,k≤d} defined by B′_{ω}(x) = U_{ω}^{∗}B_{ω}(x)U_{ω} satisfies

B_{ω,2m−1}^{′,2ℓ−1}(x) = B_{ω,2m}^{′,2ℓ}(x) and B_{ω,2m}^{′,2ℓ−1}(x) = −B_{ω,2m−1}^{′,2ℓ}(x)  (2.7)

for any 1 ≤ ℓ ≤ m ≤ h and x ∈ Λ(R_{0}), and becomes the direct sum of B_{ω}^{(ℓ)}J, ℓ = 1, 2, …, h, at x = 0, where B_{ω}^{(ℓ)} ≥ 0 and

J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .  (2.8)

(2.7) means that B′_{ω}(x) is a representation by a real matrix of a skew-Hermitian matrix 𝔹_{ω}(x) = (𝔹_{ω,m}^{ℓ}(x))_{1≤ℓ,m≤h}, where Re𝔹_{ω,m}^{ℓ}(x) = B_{ω,2m−1}^{′,2ℓ−1}(x) = B_{ω,2m}^{′,2ℓ}(x) and Im𝔹_{ω,ℓ}^{ℓ}(x) = B_{ω,2m}^{′,2ℓ−1}(x) = −B_{ω,2m−1}^{′,2ℓ}(x) for any 1 ≤ ℓ, m ≤ h. We use the condition (b-vii) in the form of (b-vii)′. This condition is always satisfied when d = 2. For the higher dimensional case, we give the following example:

Example 2.1. For a 4 × 4 real skew symmetric matrix valued random field B_{ω}(x) to be commutative with the matrix

J_{2} = ( \begin{matrix}J & O \\ O & J \end{matrix} ) ,

it is necessary and sufficient that

B_{ω}(x) = ( \begin{matrix}a_{ω}(x)J & c_{ω}(x)E + d_{ω}(x)J \\ −c_{ω}(x)E + d_{ω}(x)J & b_{ω}(x)J \end{matrix} ) ,

where E is the 2 × 2 identity matrix. For this random field to satisfy the condition (b-vii) with the complex structure J_{2}, it is necessary and sufficient that

a_{ω}(0)b_{ω}(0) ≥ c_{ω}(0)^{2} + d_{ω}(0)^{2}.  (2.9)

We define D_{ω}^{ξ}, H_{ω}^{±,ξ}, H_{ω}^{±,R,ξ}, q_{ω}^{ξ}, N_{ω}^{±,R}(λ; ξ) and N^{±}(λ; ξ) by replacing the vector potential A_{ω} by ξA_{ω} in the definitions of D_{ω}, H_{ω}^{±}, H_{ω}^{±,R}, q_{ω}, N_{ω}^{±,R}(λ) and N^{±}(λ), respectively.

Then Theorem 2 is extended as follows:

Theorem 5. Under the conditions (b-i)–(b-iii), (b-v)–(b-vii), let N^{±}(λ; ξ) be the integrated density of states of the random Pauli Hamiltonian H_{ω}^{±,ξ}. Then it holds that

lim_{ξ↑∞} \frac{N^{±}(λ; ξ)}{ξ^{h}} = E[\{(\frac{−1}{2π})^{h}Pff(B_{ω}(0))\}_{±}]  (2.10)

for each λ > 0.

Since Pff(B)^{2} = det(B), to prove Theorem 5, we have only to prove

lim_{ξ↑∞} \frac{N(λ; ξ)}{ξ^{h}} = \frac{E[\sqrt{det B_{ω}(0)}]}{(2π)^{h}},  (2.11)

where N(λ; ξ) = N^{+}(λ; ξ) + N^{−}(λ; ξ) is the integrated density of states of the Pauli Hamiltonian H_{ω}^{ξ} = H_{ω}^{+,ξ} ⊕ H_{ω}^{−,ξ}.

We prove the upper estimate in the next section and the lower estimate in Section 5. In Section 1, we use the following representation of the matrices {γ_{j}}_{j}:

γ_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and γ_{2} = ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} )  (2.12)

on V = ℂ^{2}. Then we have

D_{ω} = ( \begin{matrix}0 & A_{ω} \\ A_{ω}^{∗} & 0 \end{matrix} ) ,

where

A_{ω} = (i∂_{1} + A_{1}^{ω}(x)) + i(i∂_{2} + A_{2}^{ω}(x))  (2.13)

and A_{ω}^{∗} is its formal adjoint.

Remark 2.2. We can replace the condition (b-i) by (b-i)′ B is ℤ^{d}-stationary and ℤ^{d}-ergodic with respect to the shift in the variable x. In this condition, the random field B is called ℤ^{d}-stationary if there exist measure pre-

serving transformations {T_{a}}_{a∈ℤ^{d}} on the underlying probability space Ω such as B_{T_{a}ω}(x) = B_{ω}(x+a) for all x ∈ ℝ^{d} and a ∈ ℤ^{d}. The ℤ^{d}-stationary random field B is called ℤ^{d}-ergodic if the transformations {T_{a}}_{a∈ℤ^{d}} are ergodic (cf. [36]).

Under this condition, we replace the conditions (b-iv)–(b-vii) appropriately. Then the integrated density of states N^{±}(λ) or N^{±}(λ; ξ), λ ∈ ℝ, of the random Pauli Hamiltonian H_{ω}^{±} or H_{ω}^{±,ξ} is defined similarly, Theorem 4 is modified as

N^{+}(λ) − N^{−}(λ) = (\frac{−1}{2π})^{h} ∫_{Λ(1)} E[Pff(B_{ω}(x))]dx  (2.14)

for each λ ≥ 0, and Theorem 5 is modified as

lim_{ξ↑∞} \frac{N^{±}(λ; ξ)}{ξ^{h}} = ∫_{Λ(1)} E[\{(\frac{−1}{2π})^{h}Pff(B_{ω}(x))\}_{±}]dx  (2.15)

for each λ > 0.

The proof is reduced to that for the case of the condition (b-i) as follows. We use a probability space _{z}Ω defined by the product of the original probability space Ω and the closed box \overline{Λ(1)} with the Lebesgue measure. On this space, we define a random fields zB on ℝ^{d} by _{z}B_{(ω,y)}(x) = B_{ω}(x + y) for (ω, y) ∈ _{z}Ω and x ∈ ℝ^{d}. Then it is easy to see that this random field satisfies all conditions required in Theorems 4 and 5 except for the ergodicity in the condition (b-i). However in the following proof of Theorems 4 and 5, the ergodicity is used only in Lemma 3.1 below. That lemma is extended to the present situation by slightly modifying its proof.

3. Proof of Theorem 5: (I) Upper estimate

In this section we assume (b-i)–(b-iii) and (b-v)–(b-vi). As in [53], we use Lemma 3.1 below: for each R > 0, we define a vector potential A^{R} = (A_{ω}^{R,j}(x))_{1≤j≤d} by

A_{ω}^{R,j}(x) = \frac{−Γ(h)}{2π^{h}} ∫ ∑_{k≠j} \frac{x^{k} − y^{k}}{|x − y|^{d}} B_{ω,k}^{R,j}(y)dy,  (3.1)

where Γ(·) is the Gamma function,

B_{ω,k}^{R,j}(y) = B_{ω,k}^{j}(y)ρ(\frac{y}{R}) + \frac{1}{R}((∂_{j}ρ)(\frac{y}{R})A_{ω}^{k}(y) − (∂_{k}ρ)(\frac{y}{R})A_{ω}^{j}(y))

and ρ ∈ C_{0}^{∞}(Λ(2) → [0, 1]) such that ρ = 1 on Λ(1). Let { \atop H_{ω}^{R,ξ}} be the operator obtained by replacing A by A^{R} in the definition of H_{ω}^{R,ξ} = H_{ω}^{+,R,ξ} ⊕ H_{ω}^{−,R,ξ}. Then { \atop H_{ω}^{R,ξ}} is unitarily equivalent with H_{ω}^{R,ξ} and satisfies the following:

Lemma 3.1. Let

Ñ(\frac{t}{2}; ξ) := ∫_{0}^{∞} e^{−tλ/2}dN(λ; ξ)

and

Ñ^{R}(\frac{t}{2}; ξ) := \frac{1}{R^{d}} ∫_{Λ(R)} E[Tr[ exp ( − \frac{t}{2}{ \atop H_{ω}^{2R,ξ}})(x, x)]]dx

for any t, ξ, R > 0, where exp(−t{ \atop H_{ω}^{2R,ξ}}/2)(x, y), (t, x, y) ∈ (0, ∞)×Λ(2R)×Λ(2R) is the integral kernel of the heat semigroup exp(−t{ \atop H_{ω}^{2R,ξ}}/2) generated by { \atop H_{ω}^{2R,ξ}}/2. Then there exist finite constants c_{1}, c_{2} and c_{3} independent of t, ξ and R such that

|Ñ(\frac{t}{2}; ξ) − Ñ^{R}(\frac{t}{2}; ξ)| ≤ \frac{c_{1}}{t^{h}} exp (c_{2}(ξt)^{ν} − c_{3} \frac{R^{2}}{t} )  (3.2)

for any t, ξ, R > 0.

The proof of this lemma is same with that of Lemma 3.1 in [53]. The fundamental tool is the representation of exp(−t{ \atop H_{ω}^{2R,ξ}}/2)(x, y) by the Feynman-Kac-Itô formula: let w = (w^{1}(t), w^{2}(t), …, w^{d}(t)) be a d-dimensional Wiener process starting at 0 and M_{x}^{ξ}(t) be the solution of the ordinary differential equation

\{ \begin{matrix}\frac{d}{dt}M_{x}^{ξ}(t) = ξM_{x}^{ξ}(t)Ξ_{x}(t), \\ M_{x}^{ξ}(0) = I \end{matrix}  (3.3)

on the space End(V) of the endomorphisms on V, where

Ξ_{x}(t) := −\frac{1}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω,k}^{j}(x + w(t)).

Then exp(−t{ \atop H_{ω}^{2R,ξ}}/2)(x, y) is represented as

exp(−t{ \atop H_{ω}^{2R,ξ}}/2)(x, y)

= E^{w}[ exp ( − iξ ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{2R,j}(x + w(s))dw^{j}(s))M_{x}^{ξ}(t)

×χ{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t}|x + w(t) = y]  (3.4)

×\frac{1}{(2πt)^{h}} exp (−\frac{|x − y|^{2}}{2t} ) ,

where E^{w} is the expectation with respect to a d-dimensional Wiener process, dw^{j}(s) is the Itô stochastic differential, and, for each Borel set A in the Wiener space, χ(A) is its indicator function (cf. [12], [33], [52]). As in (3.4) of [53] we have

E^{ω}[∥M_{x}^{ξ}(t)∥_{2}^{p}] ≤ c_{1}^{p} exp(c_{2}(pξt)^{ν})  (3.5)

for any p ≥ 1, where ∥ · ∥_{2} is the Hilbert-Schmidt norm.

We now proceed to prove the upper estimate. By using

A^{∘}_{ω}^{2R,j}(x) := \frac{−Γ(h)}{2π^{h}} ∫ ∑_{k≠j} \frac{x^{k} − y^{k}}{|x − y|^{d}} \{dρ(y/(2R))A^{∘}_{ω}(y)\}_{k}^{j}dy,

A^{∘}_{ω}^{j}(x) := ∑_{k=1}^{d} B_{ω,j}^{k}(0)\frac{x^{k}}{2}

and M^{∘}^{ξ}(t) := exp(ξtΞ_{0}(0)), we decompose Ñ^{R}(t/2; ξ) as the sum of the following three terms:

Ñ_{0}^{R}(\frac{t}{2}; ξ) := ∫_{Λ(R)} \frac{dx}{R^{d}} E[ exp ( − iξ ∑_{j=1}^{d} ∫_{0}^{t} A^{∘}_{ω}^{2R,j}(x + w(s))dw^{j}(s))Tr[M^{∘}^{ξ}(t)]

×χ\{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}} ,

Ñ_{1}^{R}(\frac{t}{2}; ξ) := ∫_{Λ(R)} \frac{dx}{R^{d}} E[ exp ( − iξ ∑_{j=1}^{d} ∫_{0}^{t} A^{∘}_{ω}^{2R,j}(x + w(s))dw^{j}(s))

×Tr[M_{x}^{ξ}(t) − M^{∘}^{ξ}(t)]

×χ\{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}}

and

Ñ_{2}^{R}(\frac{t}{2}; ξ) := ∫_{Λ(R)} \frac{dx}{R^{d}} E[( exp ( − iξ ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{2R,j}(x + w(s))dw^{j}(s))

− exp ( − iξ ∑_{j=1}^{d} ∫_{0}^{t} A^{∘}_{ω}^{2R,j}(x + w(s))dw^{j}(s)))Tr[M_{x}^{ξ}(t)]

×χ\{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}} .

Since the magnetic field for A^{∘}_{ω}^{2R}(x) is constant on Λ(2R), we have

Ñ_{0}^{R}(\frac{t}{2}; ξ) = ∫_{Λ(R)} \frac{dx}{R^{d}} E[ exp (i\frac{ξ}{2} ∑_{j,k=1}^{d} B_{ω,j}^{k}(0) ∫_{0}^{t} w(s)^{k}dw^{j}(s))Tr[M^{∘}^{ξ}(t)]

×χ\{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}} .

By the same argument in the proof of Lemma 3.1, we can estimate this as

|Ñ_{0}^{R}(\frac{t}{2}; ξ) − Ñ_{0}(\frac{t}{2}; ξ)| ≤ \frac{c_{1}}{t^{h}} exp (c_{2}(ξt)^{ν} − \frac{R^{2}}{2t} ),  (3.6)

where

Ñ_{0}(\frac{t}{2}; ξ) = E[ exp (i\frac{ξ}{2} ∑_{j,k=1}^{d} B_{ω,j}^{k}(0) ∫_{0}^{t} w(s)^{k}dw^{j}(s))Tr[M^{∘}^{ξ}(t)]|w(t) = 0]\frac{1}{(2πt)^{h}} .

For each ω, there exists an orthogonal matrix U_{ω} such that U_{ω}^{∗}B_{ω}(0)U_{ω} is the direct sum of B_{ω}^{(ℓ)}J, ℓ = 1, 2, …, h, where B_{ω}^{(1)}, …, B_{ω}^{(h)} ≥ 0 and J is the matrix defined in (2.8). Since the system of the Hermitian matrices {γ̂_{k} := ∑_{j} U_{ω,k}^{j}γ_{j}} also satisfies the relation (2.2), we have

Tr[M^{∘}^{ξ}(t)] = ∏_{ℓ=1}^{h} 2 cosh (\frac{ξt}{2}B_{ω}^{(ℓ)}).

Moreover, by the O(d)-invariance of the Wiener measure and the formula for the stochastic area due to P. Lévy (cf.[33] VI-(6.10)), we have

E^{w}[ exp (i\frac{ξ}{2} ∑_{j,k=1}^{d} B_{ω,j}^{k}(0) ∫_{0}^{t} w(s)^{k}dw^{j}(s))|w(t) = 0] = ∏_{ℓ=1}^{h} \frac{ξtB_{ω}^{(ℓ)}/2}{sinh(ξtB_{ω}^{(ℓ)}/2)}.

Thus we have

\frac{Ñ_{0}(t/2; ξ)}{ξ^{h}} = E[ ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}}{2π} coth (\frac{ξtB_{ω}^{(ℓ)}}{2} )].  (3.7)

For Ñ_{1}^{R}(t/2; ξ), we use

M_{x}^{ξ}(t) − M^{∘}^{ξ}(t) = ξ ∫_{0}^{t} M_{x}^{ξ}(s)(Ξ_{x}(s) − Ξ(0))M^{∘}^{ξ}(t − s)ds

and

|Tr[M_{x}^{ξ}(t) − M^{∘}^{ξ}(t)]| ≤ ξ ∫_{0}^{t} ds∥Ξ_{x}(s) − Ξ_{0}(0)∥_{op}∥M^{∘}^{ξ}(t − s)∥_{2}∥M_{x}^{ξ}(s)∥_{2},

where ∥ · ∥_{1}, ∥ · ∥_{2} and ∥ · ∥_{op} are the trace norm, the Hilbert-Schmidt norm and the operator norm, respectively. Then we have

|Ñ_{1}^{R}(\frac{t}{2}; ξ)| ≤ \frac{c_{1}ξ}{t^{h−1}} sup_{x∈Λ(2R)} E[∥B_{ω}(x) − B_{ω}(0)∥_{2}^{3}]^{1/3} exp(c_{2}(ξt)^{ν}).  (3.8)

For Ñ_{2}^{R}(t/2; ξ), we use Lemma 2.3 in [23] (Lemma 2.2 in [42]) as follows:

E[| ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{2R,j}(x + w(s))dw^{j}(s) − ∑_{j=1}^{d} ∫_{0}^{t} A^{∘}_{ω}^{2R,j}(x + w(s))dw^{j}(s)|^{2}

×χ{x + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t}|w(t) = 0]  (3.9)

≤ E[| ∑_{j=1}^{d} ∫_{0}^{t}(A_{ω}^{2R,j} − A^{∘}_{ω}^{2R,j})(x + w(s))

×χ_{Λ(2R)}(x + w(s))dw^{j}(s)|^{2}|w(t) = 0]

≤ ct^{3/4}E[ ∫_{0}^{t} |(A_{ω}^{2R} − A^{∘}_{ω}^{2R})(x + w(s))|^{8}χ_{Λ(2R)}(x + w(s))ds|w(t) = 0]^{1/4},

where χ_{A}(x) is the indicator function on ℝ^{d} for a Borel set A in ℝ^{d}. By the definition (3.1) of the vector potential we have

E^{ω}[|(A_{ω}^{2R} − A^{∘}_{ω}^{2R})(x + w(s))|^{8}] ≤ cR^{8} sup_{y∈Λ(4R)} E[∥B_{ω}(y) − B_{ω}(0)∥_{2}^{8}].

Thus we have

|Ñ_{2}^{R}(\frac{t}{2}; ξ)| ≤ \frac{c_{1}ξR}{t^{h−1/2}} sup_{y∈Λ(4R)} E[∥B_{ω}(y) − B_{ω}(0)∥^{8}]^{1/8} exp(c_{2}(ξt)^{ν}).  (3.10)

By (3.2), (3.6), (3.7), (3.8) and (3.10), we obtain

\frac{Ñ(t/2; ξ)}{ξ^{h}} ≤ E[ ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}}{2π} coth (\frac{ξtB_{ω}^{(ℓ)}}{2} )] + \frac{c_{1}}{(ξt)^{h}} exp (c_{2}(ξt)^{ν} − \frac{R^{2}}{2t} )

+\frac{c_{3}}{(ξt)^{h−1}} (1 + \frac{R}{\sqrt{t}})B(R) exp(c_{4}(ξt)^{ν}),  (3.11)

where B(R) is a strictly increasing continuous function such that B(0) = 0 and

B(R) ≥ sup_{x∈Λ(4R)} E[∥B_{ω}(x) − B_{ω}(0)∥^{8}]^{1/8}.

The function B(R) exists because of the conditions (b-ii) and (b-v). We will take the limit as ξ → ∞, t → 0 and R → 0 so that ξt → ∞, c_{2}(ξt)^{ν} − R^{2}/(2t) = 0 and B(R) exp(c_{4}(ξt)^{ν}) = exp(−c_{2}(ξt)^{ν}), where c_{2} and c_{4} are the constants in (3.11). It is possible, for example, by setting

R = R(Ξ) = B^{−1}(exp(−(c_{2} + c_{4})Ξ^{ν})),
t = t(Ξ) = B^{−1}(exp(−(c_{2} + c_{4})Ξ^{ν}))^{2}/(2c_{2}Ξ^{ν}),
ξ = ξ(Ξ) = 2c_{2}Ξ^{ν+1}/B^{−1}(exp(−(c_{2} + c_{4})Ξ^{ν}))^{2}

and letting Ξ → ∞. Then we obtain

lim sup_{Ξ→∞} \frac{Ñ(t(Ξ)/2; ξ(Ξ))}{ξ(Ξ)^{h}} ≤ E[ ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}}{2π} ] = \frac{E[\sqrt{det B_{ω}(0)}]}{(2π)^{h}}.

Since Ñ(t/2; ξ) ≥ e^{−tλ/2}N(λ; ξ) for each λ > 0, we obtain

lim sup_{ξ↑∞} \frac{N(λ; ξ)}{ξ^{h}} ≤ \frac{E[\sqrt{det B_{ω}(0)}]}{(2π)^{h}}.

4. Proof of Theorems 4

In this section we assume the conditions (b-i)–(b-iv).

We first prove the following:

Lemma 4.1. Ñ^{+}(t) − Ñ^{−}(t) is independent of t > 0, where Ñ^{±}(t) := ∫_{0}^{∞} e^{−tλ}dN^{±}(λ).

Proof. Since the condition (b-v) is replaced by (b-iv), Lemma 3.1 is modified as follows: there exist t_{0} > 0 and c_{1}, c_{2} such that

|Ñ^{±}(\frac{t}{2}) − Ñ^{±,R}(\frac{t}{2})| ≤ \frac{c_{1}}{t^{h}} exp ( − c_{2} \frac{R^{2}}{t} )  (4.1)

for any 0 ≤ t ≤ t_{0}, where

Ñ^{±,R}(\frac{t}{2}) := \frac{1}{R^{d}} ∫_{Λ(R)} E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(x, x)Γ_{±}]]dx

and H̃_{ω}^{2R} is the operator { \atop H_{ω}^{2R,ξ}} with ξ = 1. We next define the operator Ĥ_{ω}^{4R} on L^{2}(ℝ^{d} → V) by replacing A by A^{4R} in the definition of H_{ω}, set

Ñ_{f}^{±,R}(\frac{t}{2}) = E[Tr[f exp ( − \frac{t}{2}Ĥ_{ω}^{4R})Γ_{±}f]]

for any f ∈ C(ℝ^{d} → [0, ∞)) such that ∫ f^{2}dx = 1, and show the following:

|Ñ^{±,R}(\frac{t}{2}) − Ñ_{f}^{±,R}(\frac{t}{2})| ≤ \frac{c_{3}}{t^{h}} \{ exp ( − c_{4} \frac{R^{2}}{t} ) + ∫_{Λ(R)^{c}} f^{2}(x)dx\}  (4.2)

for any 0 ≤ t ≤ t_{0}. For this we use the Feynman-Kac-Itô formula and Mercer’s expansion theorem to rewrite Ñ_{f}^{±,R}(t/2) as

∫_{ℝ^{d}} E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]]f(x)^{2}dx.

By using

Ñ^{±,R}(\frac{t}{2}) ≤ sup_{y∈Λ(R)} E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]

and the nonnegativity of Tr[exp(−tH̃_{ω}^{2R})(y, y)Γ_{±}] and Tr[exp(−tĤ_{ω}^{4R})(x, x)Γ_{±}], we estimate as

|Ñ^{±,R}(\frac{t}{2}) − Ñ_{f}^{±,R}(\frac{t}{2})|

≤ ∫ | sup_{y∈Λ(R)} E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]

−E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]]|f(x)^{2}dx

≤ sup_{x,y∈Λ(R)} |E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]] − E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]|

+( ∫_{Λ(R)^{c}} f^{2}dx)( sup_{x∈ℝ^{d}} E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]]

+ sup_{y∈Λ(R)} E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]).

As in (4.1) we have

sup_{y∈Λ(R)} E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]] ≤ \frac{c_{5}}{t^{h}}

and

sup_{x∈ℝ^{d}} E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]] ≤ \frac{c_{6}}{t^{h}}  (4.3)

for any 0 ≤ t ≤ t_{0}. In the representation of the Feynman-Kac-Itô formula, if x, y ∈ Λ(R) and y + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t, then x + w(s) ∈ Λ(4R) for any 0 ≤ s ≤ t. Thus, by the stationarity of the random magnetic field and the gauge invariance, we have

E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]

= E[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{4R,j}(x + w(s))dw^{j}(s))Tr[{ \atop M_{x}^{4R}}(t)Γ_{±}]

×χ\{y + w(s) ∈ Λ(2R) for some 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}} ,

where { \atop M_{x}^{4R}}(t) is the endomorphism valued process obtained by replacing B_{ω} by B_{ω}^{4R} in the definition (3.3) of M_{x}^{ξ}(t) with ξ = 1. Therefore we have

E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]] − E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]

= E[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{4R,j}(x + w(s))dw^{j}(s))Tr[{ \atop M_{x}^{4R}}(t)Γ_{±}]

×χ\{y + w(s) ∉ Λ(2R) for some 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{h}} .

Then, by a standard argument on the Wiener process, we have

|E[Tr[ exp ( − \frac{t}{2}Ĥ_{ω}^{4R})(x, x)Γ_{±}]] − E[Tr[ exp ( − \frac{t}{2}H̃_{ω}^{2R})(y, y)Γ_{±}]]|

≤ \frac{c_{7}}{t^{h}} exp ( − c_{8} \frac{R^{2}}{t} )

and we obtain (4.2) for any 0 ≤ t ≤ t_{0}.

We decompose { \atop N_{f}^{±,R}}(t/2) as the sum of the following three terms:

{ \atop N_{f,ε,1}^{±,R}} := E[Tr[fE([0, ε] : { \atop H_{ω}^{±,4R}})Γ_{±}f]],

{ \atop N_{f,ε,2}^{±,R}}(\frac{t}{2}) := E[Tr[f\{ exp ( − \frac{t}{2}{ \atop H_{ω}^{±,4R}}) − 1\}E([0, ε] : { \atop H_{ω}^{±,4R}})Γ_{±}f]]

and

{ \atop N_{f,ε,3}^{±,R}}(t/2) := E[Tr[f exp ( − \frac{t}{2}{ \atop H_{ω}^{±,4R}})E((ε, ∞) : { \atop H_{ω}^{±,4R}})Γ_{±}f]]

where ε ∈ (0, ∞) and E(Λ : { \atop H_{ω}^{±,4R}}), Λ ∈ B(ℝ), is the resolution of the identity of the Pauli Hamiltonian { \atop H_{ω}^{±,4R}}. It is easy to show

|{ \atop N_{f,ε,2}^{±,R}}(\frac{t}{2})| ≤ (e^{tε/2} − 1)E[Tr[f exp ( − \frac{t}{2}{ \atop H_{ω}^{±,4R}})f]] ≤ (e^{tε/2} − 1)\frac{c_{9}}{t^{h}}  (4.4)

for any 0 ≤ t ≤ t_{0}.

A key estimate is

|{ \atop N_{f,ε,3}^{+,R}}(\frac{t}{2}) − { \atop N_{f,ε,3}^{±,R}}(\frac{t}{2})| ≤ \frac{c_{10}}{εt^{h}} (1 + \frac{1}{t^{2}} + ∥∇f∥^{2})^{1/2}∥∇f∥  (4.5)

for any 0 ≤ t ≤ t_{0}, where ∥ · ∥ is the L^{2}-norm. To prove this, we use a theory on the supersymmetry (cf. [18] §6.3): let D̂_{ω}^{4R} be the Dirac operator obtained by replacing A by A^{4R} in the definition of D_{ω}. Then (Ĥ_{ω}^{4R}, Γ, D̂_{ω}^{4R}) has supersymmetry in the sense of §6.3 in [18]. As in the proof of Theorem 6.3 in [18], D̂_{ω}^{4R} is invertible on RanE((ε, ∞) : Ĥ_{ω}^{4R}) and

exp ( − \frac{t}{2}{ \atop H_{ω}^{+,4R}})E((ε, ∞) : { \atop H_{ω}^{+,4R}})Γ_{+}

= (D̂_{ω}^{4R})^{−1} exp ( − \frac{t}{2}{ \atop H_{ω}^{−,4R}})E((ε, ∞) : { \atop H_{ω}^{−,4R}})Γ_{−}D̂_{ω}^{4R}.

Since

D̂_{ω}^{4R}f^{2} − f^{2}D̂_{ω}^{4R} = 2f ∑_{j=1}^{d} γ_{j}(i∂_{j}f),

we have

{ \atop N_{f,ε,3}^{+,R}}(\frac{t}{2}) − { \atop N_{f,ε,3}^{−,R}}(\frac{t}{2})

= 2E[Tr[f(D̂_{ω}^{4R})^{−1} exp ( − \frac{t}{2}{ \atop H_{ω}^{±,4R}})E((ε, ∞) : { \atop H_{ω}^{−,4R}}) ∑_{j=1}^{d} γ_{j}(i∂_{j}f)]].

By (D̂_{ω}^{4R})^{−1} = D̂_{ω}^{4R}(Ĥ_{ω}^{4R})^{−1}, we have

|{ \atop N_{f,ε,3}^{+,R}}(\frac{t}{2}) − { \atop N_{f,ε,3}^{−,R}}(\frac{t}{2})|

≤ 2E[|||fD̂_{ω}^{4R} exp ( − \frac{t}{4}{ \atop H_{ω}^{−,4R}})E((ε, ∞) : { \atop H_{ω}^{−,4R}})|||_{2}^{2}]^{1/2}

×E[|||({ \atop H_{ω}^{−,4R}})^{−1} exp ( − \frac{t}{4}{ \atop H_{ω}^{−,4R}})E((ε, ∞) : { \atop H_{ω}^{−,4R}}) ∑_{j=1}^{d} γ_{j}(i∂_{j}f)|||_{2}^{2}]^{1/2},

where ||| · |||_{2} is the Hilbert-Schmidt norm of the operators acting on L^{2}(ℝ^{d} → V). The first factor is estimated as

E[|||fD̂_{ω}^{4R} exp ( − \frac{t}{4}{ \atop H_{ω}^{−,4R}})E((ε, ∞) : { \atop H_{ω}^{−,4R}})|||_{2}^{2}] ≤ c_{11}(1 + \frac{1}{t^{2}} + ∥∇f∥^{2}). (4.6)

In fact, for any Φ ∈ RanE((ε, ∞) : { \atop H_{ω}^{−,4R}}), we have

∥fD̂_{ω}^{4R}Φ∥_{V}^{2} = ⟨fΦ, f{ \atop H_{ω}^{−,4R}}Φ⟩_{V} + 2⟨ ∑_{j=1}^{d} γ_{j}(i∂_{j}f)Φ, fD̂_{ω}^{4R}Φ⟩_{V}

≤ \frac{1}{2}∥fΦ∥_{V}^{2} + \frac{1}{2}∥f{ \atop H_{ω}^{−,4R}}Φ∥_{V}^{2} + \frac{1}{η}∥ ∑_{j=1}^{d} γ_{j}(i∂_{j}f)Φ∥_{V}^{2} + η∥fD̂_{ω}^{4R}Φ∥_{V}^{2}

for any η ∈ (0, ∞) and

∥fD̂_{ω}^{4R}Φ∥_{V}^{2} ≤ c_{12}(∥fΦ∥_{V}^{2} + ∥f{ \atop H_{ω}^{−,4R}}Φ∥_{V}^{2} + ∥ ∑_{j=1}^{d} γ_{j}(i∂_{j}f)Φ∥_{V}^{2} ).

Then, by |||{ \atop H_{ω}^{−,4R}} exp(−t{ \atop H_{ω}^{−,4R}}/8)||| ≤ c_{13}/t and (4.3), we have (4.6). Similarly we have

E[|||({ \atop H_{ω}^{−,4R}})^{−1} exp ( − \frac{t}{4}{ \atop H_{ω}^{−,4R}}) ∑_{j=1}^{d} γ_{j}(i∂_{j}f)|||_{2}^{2}] ≤ \frac{c_{14}}{εt^{h}} ∥∇f∥^{2}.

Thus we obtain (4.5).

We take the function f as

f_{a}(x) := \frac{1}{(2πa)^{h/2}} exp ( − \frac{|x|^{2}}{4a} )

for a > 0. Then we have

∫_{Λ(R)^{c}} f_{a}^{2}(x)dx ≤ c_{15} exp ( − c_{16} \frac{R^{2}}{a} ).

To obtain the decay in (4.2) as R → ∞, we take a as a(R) = R^{2−α} for α > 0. Then we have ∥∇f_{a(R)}∥ = \sqrt{d}/(2R^{1−α/2}). To obtain the decay in (4.4) and (4.5) as R → ∞, we take ε as ε(R) = R^{α/2+β−1} and take α, β > 0 so that α/2 + β < 1. Then we obtain

Ñ^{+}(\frac{t}{2}) − Ñ^{−}(\frac{t}{2}) = lim_{R→∞}(Ñ_{f_{a(R)},ε(R),1}^{+,R} − Ñ_{f_{a(R)},ε(R),1}^{−,R})

for any 0 ≤ t ≤ t_{0}. The right hand side is independent of t. By the theorem of the identity of analytic functions, Ñ^{+}(t) − Ñ^{−}(t) is independent of all t > 0.

Remark 4.1. If we replace the condition (b-ii) by

(b-ii)′ the sample path x ↦ B_{ω}(x) is differentiable, then we can give a simple proof of Lemma 4.1: in this case, the vector potential A_{ω} by the Poincaré gauge is differentiable and the integral kernel exp(−tH_{ω}/2)(x, y) is represented by the Feynman-Kac-Itô formula. Then we can show N^{±}(λ) = E[Tr[fE([0, λ] : H_{ω}^{±})f]] for any f ∈ C(ℝ^{d} → [0, ∞)) such that ∫ f^{2}dx = 1 (cf. [14] Proposition VI.1.3). As in the proof of (4.5), we have

|(N^{+}(λ) − N^{−}(λ)) − (N^{+}(ε) − N^{−}(ε))|

= |E[Tr[fE((ε, λ] : H_{ω}^{+})f]] − E[Tr[fE((ε, λ] : H_{ω}^{−})f]]|

= |2E[Tr[fD_{ω}^{−1}E((ε, λ] : H_{ω}^{−}) ∑_{j=1}^{d} γ_{j}(i∂_{j}f)]]|

≤ \frac{c}{ε}(1 + λ^{2} + ∥∇f∥^{2})^{1/2}∥∇f∥

for any 0 < ε < λ. Since f is arbitrary, N^{+}(λ) − N^{−}(λ) is independent of λ ≥ 0 and Ñ^{+}(t) − Ñ^{−}(t) is independent of t ≥ 0.

Proof of Theorem 4. The rest of the proof is only to show

lim_{t↓0} \{Ñ^{+}(\frac{t}{2}) − Ñ^{−}(\frac{t}{2})\} = (\frac{−1}{2π})^{h}E[Pff(B_{ω}(0))].

For this we use the heat equation method to prove the index theorem (cf. [18] §12). By (4.1), it is sufficient to show

lim_{t↓0} \{{ \atop N^{+,1}}(\frac{t}{2}) − { \atop N^{−,1}}(\frac{t}{2})\} = (\frac{−1}{2π})^{h}E[Pff(B_{ω}(0))].  (4.7)

By the scaling property of the Wiener process, we have

{ \atop N^{+,1}}(\frac{t}{2}) − { \atop N^{−,1}}(\frac{t}{2})

= ∫_{Λ(1)} dxE[ exp ( − i\sqrt{t} ∑_{j=1}^{d} ∫_{0}^{1} A_{ω}^{2,j}(x + \sqrt{t}w(s))dw^{j}(s))Str[M_{x}^{(t)}(1)] (4.8)

×χ\{x + \sqrt{t}w(s) ∈ Λ(2) for any 0 ≤ s ≤ 1\}|w(1) = 0]\frac{1}{(2πt)^{h}} ,

where, for any M ∈ End(V), Str[M] = Tr[MΓ] is the supertrace, {M_{x}^{(t)}(s) : s ≥ 0} is the endomorphism valued process obtained by replacing Ξ_{x}(s) and ξ by

Ξ_{x}^{(t)}(s) := −\frac{1}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω,k}^{j}(x + \sqrt{t}w(s))

and t, respectively, in the ordinary differential equation (3.3). By using the ordinary differential equation, we have

M_{x}^{(t)}(1) = I + tM_{1} + t^{2}M_{2} + ⋯ + t^{n}M_{n} + t^{n+1}M_{n,R}

for any n ∈ ℕ, where

M_{p} = ∫ ⋯ ∫_{0≤s_{p}≤···≤s_{2}≤s_{1}≤1} Ξ_{x}^{(t)}(s_{p}) ⋯ Ξ_{x}^{(t)}(s_{2})Ξ_{x}^{(t)}(s_{1})ds_{p} ⋯ ds_{2}ds_{1}

for p = 1, 2, …, n, and

M_{n,R} = ∫ ⋯ ∫_{0≤s_{n+1}≤···≤s_{2}≤s_{1}≤1} M_{x}^{(t)}(s_{n+1})Ξ_{x}^{(t)}(s_{n+1}) ⋯ Ξ_{x}^{(t)}(s_{2})Ξ_{x}^{(t)}(s_{1})ds_{n+1} ⋯ ds_{2}ds_{1}.

We now use the Berezin formula:

Str[ ∑_{p=0}^{d} ∑_{1≤j_{1}<j_{2}<···<j_{p}≤d} C_{j_{1}j_{2}···j_{p}}γ_{j_{1}}γ_{j_{2}} ⋯ γ_{j_{p}}] = (−2i)^{h}C_{12···d}  (4.9)

(cf. [18] §12.2). Then we have Str[M_{p}] = 0 for any p < h. Therefore we take n = h. Since E^{ω}[∥M_{x}^{(t)}(s)∥_{2}^{2}] ≤ c for 0 ≤ t ≤ t_{0} and 0 ≤ s ≤ 1 as in (3.5), we can show that the function obtained by replacing M_{x}^{(t)}(1) by t^{h+1}M_{h,R} in (4.8) tends to zero as t → 0. Moreover, by Lemma 2.3 in [23] (Lemma 2.2 in [42]), we have

sup_{0<t≤1} E[| ∑_{j=1}^{d} ∫_{0}^{1} A_{ω}^{2,j}(x + \sqrt{t}w(s))dw^{j}(s)|^{2}|w(1) = 0] < ∞.

Therefore the right hand side of (4.7) equals

lim_{t↓0} ∫_{Λ(1)} dxE[Str[M_{h}]χ\{x + \sqrt{t}w(s) ∈ Λ(2) for any 0 ≤ s ≤ 1\}|w(1) = 0]\frac{1}{(2π)^{h}} .

Since the integrand is uniformly integrable, this equals

E[Str[( − \frac{1}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω,k}^{j}(0))^{h}]]\frac{1}{h!(2π)^{h}} .

By using again the Berezin formula (4.9), we can show that this coincides with the right hand side of (4.7).

5. Proof of Theorem 5: (II) Lower estimate

In this section we assume the conditions (b-i)–(b-iii) and (b-v)–(b-vii). By the orthogonal matrix U_{ω} in the condition (b-vii)′, we define a distance on ℝ^{d} by

d_{ω}(x, y) := max_{1≤ℓ≤h} ( ∑_{j=2ℓ−1}^{2ℓ}{∑_{k=1}^{d} U_{ω,j}^{k}(x^{k} − y^{k})}^{2})^{1/2}

for any x, y ∈ ℝ^{d}. For each p ∈ ℝ^{d} and r > 0, we define a polydisk by D_{ω}(p, r) := {x ∈ ℝ^{d} : d_{ω}(p, x) < r}. We take an osculatory packing of the unit cube Λ(1) by the polydisks inductively so that the related quantities are measurable as follows: we first set r_{1}^{ω} := sup{r > 0 : D_{ω}(p, r) ⊂ Λ(1) for some p ∈ Λ(1)}, which is measurable in ω. Then the set S^{ω}(1, 0) := {p ∈ Λ(1) : D_{ω}(p, r_{1}^{ω}) ⊂ Λ(1)} is not empty. We next set p_{1}^{ω,1} := sup{p^{1} : p ∈ S^{ω}(1, 0)} and S^{ω}(1, 1) := {p ∈ S^{ω}(1, 0) : p^{1} = p_{1}^{ω,1}}. Then p_{1}^{ω,1} is measurable in ω and S^{ω}(1, 1) is not empty. Similarly, when the numbers p_{1}^{ω,1}, p_{1}^{ω,2}, …, p_{1}^{ω,j−1} and the nonempty sets S^{ω}(1, 0), S^{ω}(1, 1), …, S^{ω}(1, j − 1) are obtained for some j ≤ d, we set p_{1}^{ω,j} := sup{p^{j} : p ∈ S^{ω}(1, j − 1)} and S^{ω}(1, j) := {p ∈ S^{ω}(1, j − 1) : p^{j} = p_{1}^{ω,j}}. Then p_{1}^{ω,j} is measurable in ω and S^{ω}(1, j) is not empty. By continuing this procedure, we obtain a Λ(1)-valued random variable p_{1}^{ω} = (p_{1}^{ω,1}, p_{1}^{ω,2}, …, p_{1}^{ω,d}). Similarly when Λ(1)-valued random variables p_{1}^{ω}, p_{2}^{ω}, …, p_{ν−1}^{ω} are obtained for some ν ≥ 2, we set

r_{ν}^{ω} := sup \{r > 0 : D_{ω}(p, r) ⊂ Λ(1) − ⋃_{μ=1}^{ν−1} \overline{D_{μ}^{ω}} for some p ∈ Λ(1)\}

and

S^{ω}(ν, 0) := \{p ∈ Λ(1) : D_{ω}(p, r_{ν}^{ω}) ⊂ Λ(1) − ⋃_{μ=1}^{ν−1} \overline{D_{μ}^{ω}}\},

where \overline{D_{μ}^{ω}} is the closure of the domain D_{μ}^{ω} := D_{ω}(p_{μ}^{ω}, r_{μ}^{ω}). Moreover, when the positive numbers p_{ν}^{ω,1}, p_{ν}^{ω,2}, …, p_{ν}^{ω,j−1} and the nonempty sets S^{ω}(ν, 0), S^{ω}(ν, 1), …, S^{ω}(ν, j − 1) are obtained for some j ≤ d, we set p_{ν}^{ω,j} := sup{p^{j} : p ∈ S^{ω}(ν, j − 1)} and S^{ω}(ν, j) := {p ∈ S^{ω}(ν, j − 1) : p^{j} = p_{ν}^{ω,j}}. By continuing this procedure, we obtain the decreasing sequence of positive random variables {r_{ν}^{ω}}_{ν=1}^{∞} and Λ(1)-valued random variables {p_{ν}}_{ν=1}^{∞} so that the sets {D_{ν}^{ω}}_{ν=1}^{∞} are mutually disjoint and is included in Λ(1). As in the usual proof of Vitali’s covering theorem, we can show that this packing is a complete packing:

∑_{ν=1}^{∞} |D_{ν}^{ω}| = 1 = |Λ(1)|,  (5.1)

where | · | is the volume.

In the following we use the ω-dependent complex coordinate (z_{ω}^{1}, …, z_{ω}^{h}) defined by z_{ω}^{ℓ} = x_{ω}^{2ℓ−1}+ix_{ω}^{2ℓ} and x_{ω}^{j} = ∑_{k=1}^{d} U_{ω,j}^{k}x^{k}. In terms of this coordinate, the polydisk D_{ω}(p, r) is the usual polydisk {z_{ω} ∈ ℂ^{h} : |z_{ω}^{ℓ} − z_{ω}^{ℓ}(p)| < r for ℓ = 1, 2, …, h} and the cube Λ(1) is a domain obtained by rotating (−1/2, 1/2)^{d}. On the other hand. the magnetic field is represented as the complex 2-form

𝔹_{ω}(z_{ω}) = \frac{1}{2} ∑_{ℓ,m} 𝔹_{ω,m}^{ℓ}(z_{ω})dz_{ω}^{ℓ} ∧ d\overline{z_{ω}^{m}}

and the vector potential defined in (2.1) is represented as the complex 1-form

𝔸_{ω}(z_{ω}) = ∑_{ℓ=1}^{h}(𝔸_{ω}^{ℓ}(z_{ω})d\overline{z_{ω}^{ℓ}} + \overline{𝔸_{ω}^{ℓ}(z_{ω})}dz_{ω}^{ℓ})

on Λ(R_{0}), where R_{0} and (𝔹_{ω,m}^{ℓ}(z_{ω}))_{1≤ℓ,m≤h} are the positive number and the skew Hermitian matrix valued random field, respectively, defined in Section 2, and

𝔸_{ω}^{ℓ}(z_{ω}) = \frac{1}{2} ∫_{0}^{1} ∑_{m=1}^{h} 𝔹_{ω,ℓ}^{m}(tz_{ω})tz_{ω}^{m}dt.

The Hermitian matrices γ_{ω,1}, γ_{ω,2}, …, γ_{ω,d} defined by γ_{ω,j} = ∑_{k=1}^{d} U_{ω,j}^{k}γ_{k} also satisfies the commutation relation (2.2). We represent these matrices as γ_{ω,2ℓ−1} = ext(δ_{ℓ}) + int(δ_{ℓ}), γ_{ω,2ℓ} = i(ext(δ_{ℓ}) − int(δ_{ℓ})) for ℓ = 1, 2, …, h, on V = Λ(ℂ^{h}), where {δ_{1}, δ_{2}, …, δ_{h}} is a unitary basis of ℂ^{h}, ext is the exterior multiplication, i.e., ext(δ_{ℓ})ψ = δ_{ℓ} ∧ ψ and int is the interior multiplication, i.e., the adjoint of ext. Then the Dirac operator is represented as

D_{ω}^{ξ} = ∑_{ℓ=1}^{h} \{ext(δ_{ℓ})2(i\frac{∂}{∂\overline{z_{ω}^{ℓ}}} + ξ𝔸_{ω}^{ℓ}(z_{ω})) + int(δ_{ℓ})2(i\frac{∂}{∂z_{ω}^{ℓ}} + ξ\overline{𝔸_{ω}^{ℓ}(z_{ω})})\}  (5.2)

on Λ(R_{0}).

Until (5.12) below we fix ω and omit to indicate the ω-dependence. For each ν = 1, 2, … and 0 < R < R_{0}, we introduce a vector potential

𝔸^{ν,R}(z) = ∑_{ℓ=1}^{h}(𝔸^{ν,R,ℓ}(z)d\overline{z^{ℓ}} + \overline{𝔸^{ν,R,ℓ}(z)}dz^{ℓ})

by 𝔸^{ν,R,ℓ}(z) := 𝔸^{∘}^{ν,R,ℓ}(z) + 𝔸^{Δ,ℓ}(z), where

𝔸^{∘}^{ν,R,ℓ}(z) := iB^{(ℓ)}(z^{ℓ} − Rp_{ν}^{ℓ})/4,

𝔸^{Δ,ℓ}(z) := \frac{1}{2} ∫_{0}^{1} ∑_{m=1}^{h} 𝔹_{ℓ}^{Δ,m}(tz)tz^{m}dt

and 𝔹^{Δ}(z) := 𝔹(z) − 𝔹(0). Let D̂^{ν,R,ξ} be the Dirac operator obtained by replacing 𝔸^{ℓ} by 𝔸^{ν,R,ℓ} in the definition of the operator D^{ξ}. Let H^{ν,R,ξ} be the self-adjoint operator on the space L^{2}(RD_{ν} → V) corresponding to the closure of the quadratic form q^{ξ}(Φ, Ψ) with the domain C_{0}^{∞}(RD_{ν} → V) and Ĥ^{ν,R,ξ} be the operator obtained by replacing D^{ξ} by D̂^{ν,R,ξ} in the definition of the operator H^{ν,R,ξ}. Then H^{ν,R,ξ} and Ĥ^{ν,R,ξ} are unitarily equivalent by the gauge invariance.

We use a uniform estimate of the solution of the \overline{∂}-equation by Berndtsson [9]:

Lemma 5.1. Let B(r) := {z ∈ ℂ^{h} : |z| < r} for r > 0. Then, for any R > 0 such that 2R\sqrt{d} < R_{0}, there exists a function ψ_{R} on B(R\sqrt{d}) such that

\overline{∂}ψ_{R} = i ∑_{ℓ=1}^{h} 𝔸^{Δ,ℓ}(z)d\overline{z^{ℓ}}  (5.3)

and

sup_{z∈Λ(R)} |ψ_{R}(z)| ≤ C_{d}R^{2} sup_{z∈B(R\sqrt{d})} max_{1≤ℓ,m≤h} |𝔹_{m}^{Δ,ℓ}(z)|,

where C_{d} depends only on the dimension.

Proof. By Theorem 2 in Berndtsson [9], there exists a function Ψ_{R} on B(1) such that

\overline{∂}Ψ_{R} = i ∑_{ℓ=1}^{h} 𝔸^{Δ,ℓ}(R\sqrt{d}z)d\overline{z^{ℓ}}

on B(1) and

sup_{z∈B(1/2)} |Ψ_{R}(z)| ≤ C′_{d} sup_{x∈B(1)} max_{1≤ℓ≤h} |𝔸^{Δ,ℓ}(R\sqrt{d}z)|.

We set ψ_{R}(z) := R\sqrt{d}Ψ_{R}(z/(R\sqrt{d})). Then (5.3) holds and

sup_{z∈B(R\sqrt{d}/2)} |ψ_{R}(z)| ≤ C′_{d}R\sqrt{d} sup_{z∈B(R\sqrt{d})} max_{1≤ℓ≤h} |𝔸^{Δ,ℓ}(z)|

≤ C_{d}R^{2} sup_{z∈B(R\sqrt{d})} max_{1≤ℓ,m≤h} |𝔹_{m}^{Δ,ℓ}(z)|.

We now take R > 0 so that 2R\sqrt{d} < R_{0}, and set

ψ_{ν,R}(z) := − ∑_{ℓ=1}^{h} \frac{B^{(ℓ)}}{4} |z^{ℓ} − Rp_{ν}^{ℓ}|^{2} + ψ_{R}(z)

and ϕ_{ν,R}(z) := exp(ξψ_{ν,R}(z)). Then we have

D̂^{ν,R,ξ}( ∏_{ℓ=1}^{h}(z^{ℓ} − Rp_{ν}^{ℓ})^{n(ℓ)})ϕ_{ν,R}(z) = 0

for any n = (n(1), n(2), ⋯, n(h)) ∈ ℤ_{+}^{h}. To obtain functions in the domain of Ĥ^{ν,R,ξ}, we introduce ζ ∈ C_{0}^{∞}(ℂ → [0, 1]) such that

ζ(z) = \{ \begin{matrix}0 if z ∉ B^{2}(1), \\ 1 if z ∈ B^{2}(1 − δ) \end{matrix}  (5.4)

and |∇ζ| ≤ c/δ, where B^{2}(r) := {x ∈ ℂ : |z| < r} for r > 0, 0 < δ < 1/2 is an arbitrary small constant and c is a constant independent of δ. For any n = (n(1), n(2), ⋯, n(h)) ∈ ℤ_{+}^{h}, we set

ϕ_{ν,n}(z) := ( ∏_{ℓ=1}^{h} ζ(\frac{z^{ℓ} − Rp_{ν}^{ℓ}}{Rr_{ν}} )(z^{ℓ} − Rp_{ν}^{ℓ})^{n(ℓ)})ϕ_{ν,R}(z).

Then we have

∥D̂^{ν,R,ξ}ϕ_{ν,n}∥^{2} ≤ \frac{c}{(Rr_{ν}δ)^{2}} ∑_{ℓ=1}^{h} ∫_{B^{2}(Rr_{ν})−B^{2}(Rr_{ν}(1−δ))} |z^{1}|^{2n(ℓ)} exp(−ξB^{(ℓ)}|z^{1}|^{2}/2)dx^{1}dx^{2}

×( ∏_{m≠ℓ} ∫ ζ^{2}(\frac{z^{1}}{Rr_{ν}} )|z^{1}|^{2n(m)} exp(−ξB^{(m)}|z^{1}|^{2}/2)dx^{1}dx^{2})

× exp (2ξ sup_{z∈Λ(R)} |ψ_{R}(z)|)

and

\frac{∥D̂^{ν,R,ξ}ϕ_{ν,n}∥^{2}}{∥ϕ_{ν,n}∥^{2}} ≤ \frac{c}{(Rr_{ν}δ)^{2}} exp (4ξ sup_{z∈Λ(R)} |ψ_{R}(z)|)

× ∑_{ℓ=1}^{h} ∫_{B^{2}(Rr_{ν})−B^{2}(Rr_{ν}(1−δ))} |z^{1}|^{2n(ℓ)} exp(−ξB^{(ℓ)}|z^{1}|^{2}/2)dx^{1}dx^{2}

×{ ∫_{B^{2}(Rr_{ν}(1−δ))} |z^{1}|^{2n(ℓ)} exp(−ξB^{(ℓ)}|z^{1}|^{2}/2)dx^{1}dx^{2}}^{−1}.

We now assume

sup_{z∈B(R\sqrt{d})} max_{1≤ℓ,m≤h} |B_{m}^{Δ,ℓ}(z)| < η.

Then we have

\frac{∥D̂^{ν,R,ξ}ϕ_{ν,n}∥^{2}}{∥ϕ_{ν,n}∥^{2}} ≤ \frac{c_{1}}{(Rr_{ν})^{2}δ} ∑_{ℓ=1}^{h} \{(1 − δ)^{2n(ℓ)}I(n(ℓ), ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}/2) exp(−ξc_{2}ηR^{2})\}^{−1},

where

I(n, a) := ∫_{0}^{1} s^{n} exp(a(1 − s))ds  (5.5)

for n, a ≥ 0.

We fix λ > 0 arbitrarily. For

∥D̂^{ν,R,ξ}ϕ_{ν,n}∥^{2}/∥ϕ_{ν,n}∥^{2} ≤ λ  (5.6)

to hold, it is sufficient that

\frac{c_{1}h}{(Rr_{ν})^{2}δλ(1 − δ)^{2n(ℓ)}} ≤ I(n(ℓ), ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}/2) exp(−ξc_{2}ηR^{2}),  (5.7)

for all ℓ ∈ {1, 2, …, h}. Introducing 0 < σ < 1/2, we use the estimate

I(n, a) ≥ \{ \begin{matrix}σ(n/(ae))^{n} exp(a(1 − σ)) if σ + n/a ≤ 1, \\ σ(1 − σ)^{n} if σ + n/a ≥ 1. \end{matrix}  (5.8)

Then, for (5.7) to hold, it is sufficient that

\frac{c_{1}h}{(Rr_{ν})^{2}δλσ} (\frac{ξB^{(ℓ)}(Rr_{ν})^{2}e}{2n(ℓ)})^{n(ℓ)} ≤ exp(ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}(1 − σ)/2 − ξc_{2}ηR^{2}) (5.9)

when n(ℓ) ≤ ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}(1 − σ)/2, and

\frac{c_{1}h}{(Rr_{ν})^{2}δλσ((1 − δ)^{2}(1 − σ))^{n(ℓ)}} ≤ exp(−ξc_{2}ηR^{2})  (5.10)

when n(ℓ) ≥ ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}(1 − σ)/2. However (5.10) is impossible for sufficiently large ξ. Therefore we consider only (5.9). This is rewritten as g_{ℓ}^{ν}(n(ℓ)) ≤ 0 and n(ℓ) ≤ ξB^{(ℓ)}(Rr_{ν}(1 − δ))^{2}(1 − σ)/2, where g_{ℓ}^{ν}(n) := −n log n + nA_{ℓ}^{ν} − B_{ℓ}^{ν} for n > 0, A_{ℓ}^{ν} := log(ξB^{(ℓ)}(Rr_{ν})^{2}e/2) and B_{ℓ}^{ν} := ξB^{(ℓ)}(Rr_{ν}(1−δ))^{2}(1−σ)/2−ξc_{2}ηR^{2}+log((Rr_{ν})^{2}δλσ/(c_{1}h)). The function g_{ℓ}^{ν}(n) is increasing on (0, exp(A_{ℓ}^{ν} − 1)] and its maximum is

g_{ℓ}^{ν}(exp(A_{ℓ}^{ν} − 1)) = exp(A_{ℓ}^{ν} − 1) − B_{ℓ}^{ν}
= ξB^{(ℓ)}(0)(Rr_{ν})^{2}(1 − (1 − δ)^{2}(1 − σ))/2 + ξc_{2}ηR^{2} − log((Rr_{ν})^{2}δλσ/(c_{1}h)).

We now assume η < (r_{ν})^{2}B^{(ℓ)}(1 − (1 − δ)^{2}(1 − σ))/(2c_{2}). Since δ, σ < 1/2, by taking ξ large enough, we may regard that lim_{n↓0} g_{ℓ}^{ν}(n) < 0 and g_{ℓ}^{ν}(exp(A_{ℓ}^{ν} − 1)) > 0. We put

k_{ℓ}^{ν,ξ} := inf{k > 1 : g_{ℓ}^{ν}(exp(A_{ℓ}^{ν} − 1)/k) ≤ 0}  (5.11)

and take another small ϑ > 0. Then g_{ℓ}^{ν}(n) < 0 for n ≤ exp(A_{ℓ}^{ν} − 1)/(k_{ℓ}^{ν,ξ} + ϑ). For n ≥ exp(A_{ℓ}^{ν} − 1)/(k_{ℓ}^{ν,ξ} + ϑ), we use log n ≥ A_{ℓ}^{ν} − 1 − log(k_{ℓ}^{ν,ξ} + ϑ) to estimate as g_{ℓ}^{ν}(n) ≤ n(1 + log(k_{ℓ}^{ν,ξ} + ϑ)) − B_{ℓ}^{ν}. Therefore, for g_{ℓ}^{ν}(n(ℓ)) ≤ 0 to hold, it is sufficient that

n(ℓ) ≤ B_{ℓ}^{ν}/(1 + log(k_{ℓ}^{ν,ξ} + ϑ)).  (5.12)

The right hand side is less than ξB^{(ℓ)}(0)(Rr_{ν}(1 − δ))^{2}(1 − σ)/2. Consequently, for (5.6) to hold, it is sufficient that (5.12) holds for all ℓ ∈ {1, 2, …, h}.

Thus the counting function N_{ω}(λ; ξ, RD_{ν}^{ω}) of the eigenvalues of the operator H_{ω}^{ν,R,ξ} is greater than or equal to

∏_{ℓ=1}^{h} (\frac{B_{ℓ,ω}^{ν}}{1 + log(k_{ℓ,ω}^{ν,ξ} + ϑ)} − 1).

On the other hand, k_{ℓ,ω}^{ν,ξ} in (5.11) is rewritten as

k_{ℓ,ω}^{ν,ξ} = K((1 − δ)^{2}(1 − σ) − \frac{2c_{2}η}{B_{ω}^{(ℓ)}(r_{ν}^{ω})^{2}} + \frac{2}{ξB_{ω}^{(ℓ)}(Rr_{ν}^{ω})^{2}} log \frac{(Rr_{ν}^{ω})^{2}δλσ}{c_{1}h} ),  (5.13)

where K is the inverse of the function F (k) = (1 + log k)/k, k > 1. Since F is strictly decreasing continuous function and lim_{k↓1} F (k) = 1, K is also strictly decreasing continuous function on the interval (0, 1) and lim_{f↑1} K(f) = 1. Therefore we obtain

lim inf_{ξ↑∞} \frac{N_{ω}(λ; ξ, RD_{ν}^{ω})}{ξ^{h}} ≥ ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}(Rr_{ν}^{ω}(1 − δ))^{2}(1 − σ)/2 − c_{2}ηR^{2}}{1 + log K((1 − δ)^{2}(1 − σ) − 2c_{2}η/(B_{ω}^{(ℓ)}(r_{ν}^{ω})^{2}))}

for almost all ω. We take another small υ > 0 and set μ_{ω}(υ) := max{μ : r_{μ}^{ω} ≥ υ}. By the min-max principle, we have

N_{ω}^{R}(λ; ξ) ≥ ∑_{ν=1}^{μ_{ω}(υ)} N_{ω}(λ; ξ, RD_{ν}^{ω}).

Thus we have

lim inf_{ξ↑∞} \frac{N_{ω}^{R}(λ; ξ)}{ξ^{h}R^{d}} ≥ ∑_{ν=1}^{μ_{ω}(υ)} |D_{ν}^{ω}| ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}(1 − δ)^{2}(1 − σ)/(2π) − c_{2}η/(πυ^{2})}{1 + log K((1 − δ)^{2}(1 − σ) − c_{2}η/(B_{ω}^{(ℓ)}υ^{2}))}.

We now take the expectation in ω. By the Akcoglu-Krengel superadditive ergodic theorem (cf. [3], [14], [37]) and Fatou’s lemma, we have

lim inf_{ξ↑∞} \frac{N(λ; ξ)}{ξ^{h}} ≥ E[ ∑_{ν=1}^{μ_{ω}(υ)} |D_{ν}^{ω}| ∏_{ℓ=1}^{h} \frac{B_{ω}^{(ℓ)}(1 − δ)^{2}(1 − σ)/(2π) − c_{2}η/(πυ^{2})}{1 + log K((1 − δ)^{2}(1 − σ) − c_{2}η/(B_{ω}^{(ℓ)}υ^{2}))}

: B_{ω}^{(1)}, …, B_{ω}^{(h)} > ε, sup_{z∈B(R\sqrt{d})} max_{1≤ℓ,m≤h} |B_{ω,m}^{Δ,ℓ}(z)| < η],

where ε > 0 is arbitrary and 0 < η < ευ^{2}(1 − (1 − δ)^{2}(1 − σ))/(2c_{2}). By taking the limit as R, η, σ, δ, ε, υ → 0, we obtain

lim inf_{ξ↑∞} \frac{N(λ; ξ)}{ξ^{h}} ≥ \frac{E[\sqrt{det B_{ω}(0)}]}{(2π)^{h}}.

6. Proof of Theorem 3

We first consider the case that the conditions (a-i), (a-ii) and (a-vi) are satisfied. Setting

C_{ω}^{j}(x^{j}) = ∫_{0}^{x^{j}} dt ∫_{0}^{t} dsB_{ω}^{j}(s),

we take the following vector potential:

A_{ω}^{j}(x) = (−1)^{j} \frac{d}{dx^{o(j)}} C_{ω}^{o(j)}(x^{o(j)}) = (−1)^{j} ∫_{0}^{x^{o(j)}} dtB_{ω}^{o(j)}(t)

for j = 1, 2, where o(1) = 2 and o(2) = 1. Then the functions ϕ_{n}^{ω}(x) = z^{n} exp(−C_{ω}^{1}(x^{1}) − C_{ω}^{2}(x^{2})), n = 0, 1, 2, …, satisfy A_{ω}ϕ_{n}^{ω} = 0, where z = x^{1} + ix^{2} and A_{ω} is the operator defined in (2.13). By a family of functions {ζ_{R}}_{R>1} ⊂ C_{0}^{∞}(ℝ → [0, 1]) satisfying

ζ_{R}(t) = \{ \begin{matrix}0 if |t| ≥ R + 1, \\ 1 if |t| ≤ R, \end{matrix}  (6.1)

and sup_{R,t} |ζ′_{R}(t)| < ∞, we restrict ϕ_{n}^{ω} as ϕ_{n,R}^{ω}(x) = ζ_{R}(x^{1})ζ_{R}(x^{2})ϕ_{n}^{ω}(x). Then we have

∥A_{ω}ϕ_{n,R}^{ω}∥^{2} ≤ c_{1} ∑_{j=1}^{2} ∫_{R≤|x^{j}|≤R+1} dx^{j} exp(−2C_{ω}^{j}(x^{j}))

× ∫ dx^{o(j)}ζ_{R}(x^{o(j)})^{2} exp(−2C_{ω}^{o(j)}(x^{o(j)}))|x|^{2n}.

When R ≤ |x^{j}| ≤ R + 1 and |x^{o(j)}| ≤ R + 1, we have |x|^{2} ≤ 5(x^{j})^{2}. Using also |x|^{2} ≥ (x^{j})^{2} for ∥ϕ_{n,R}^{ω}∥^{2}, we have

\frac{∥A_{ω}ϕ_{n,R}^{ω}∥^{2}}{∥ϕ_{n,R}^{ω}∥^{2}} ≤ c_{1}5^{n} ∑_{j=1}^{2} ∫_{R≤|t|≤R+1} t^{2n} exp(−2C_{ω}^{j}(t))dt/ ∫_{|t|≤R} t^{2n} exp(−2C_{ω}^{j}(t))dt.

By the uniform estimate, we have

∫_{R≤|t|≤R+1} t^{2n} exp(−2C_{ω}^{j}(t))dt ≤ 2(R + 1)^{2n} exp(−2 inf_{R≤|t|≤R+1} C_{ω}^{j}(t)).

Moreover, by restricting the integral to ηR/2 ≤ |t| ≤ ηR, we have

∫_{|t|≤R} t^{2n} exp(−2C_{ω}^{j}(t))dt ≥ \frac{(ηR)^{2n+1}}{2n + 1} exp ( − 2 sup_{ηR/2≤|t|≤ηR} C_{ω}^{j}(t)),

where 0 < η < 1 is specified later. To estimate these quantities, we use the process D_{ω}^{j}(t) := C_{ω}^{j}(t)/t^{2} instead of C_{ω}^{j}(t), since the covariance of D_{ω}^{j}(t) is bounded in t.

We now estimate the probability

p_{j}(a_{1}, a_{2}) := P ( inf_{R≤|t|≤R+1} D_{ω}^{j}(t) ≥ a_{1} and sup_{ηR/2≤|t|≤ηR} D_{ω}^{j}(t) ≤ a_{2})

from below for any a_{1}, a_{2} ≥ 0. Let D_{ω}^{j,k}(t) = (D_{ω}^{j}(t) + (−1)^{k}D_{ω}^{j}(−t))/2, k = 1, 2, be the even and odd parts of D_{ω}^{j}(t), respectively. Then we have

p_{j}(a_{1}, a_{2}) ≥ P ( inf_{R≤t≤R+1} D_{ω}^{j,1}(t) ≥ a_{1} + a_{3} and sup_{ηR/2≤t≤ηR} D_{ω}^{j,1}(t) ≤ a_{2} − a_{3})

×P ( sup_{ηR/2≤t≤ηR or R≤t≤R+1} |D_{ω}^{j,2}(t)| ≤ a_{3})

for any a_{3} ≥ 0, since D_{ω}^{j,1} and D_{ω}^{j,2} are independent as random processes. We decompose

the process D_{ω}^{j,1} as the sum of D^{∘}_{ω}^{j,1}(t) = E[D_{ω}^{j,1}(t)|D_{ω}^{j,1}(R)] and \overline{D_{ω}^{j,1}}(t) = D_{ω}^{j,1}(t)−D^{∘}_{ω}^{j,1}(t).

These processes are independent and D^{∘}_{ω}^{j,1}(t) = X^{j}(t)W_{ω}^{j}, where W_{ω}^{j} is a random variable obeying the standard normal distribution and X^{j}(t) = E[D_{ω}^{j,1}(t)D_{ω}^{j,1}(R)]/E[D_{ω}^{j,1}(R)^{2}]^{1/2}.

In the following we assume 0 < r < 1/3. This is always possible under the condition (a-vi). By the representation

E[D_{ω}^{j,1}(t)D_{ω}^{j,1}(s)]  (6.2)

= \frac{1}{2} ∫_{0}^{1} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{0}^{1} dv_{1} ∫_{0}^{v_{1}} dv_{2}\{β^{j}(tu_{2} − sv_{2}) + β^{j}(tu_{2} + sv_{2})\}

and the condition inf_{|x^{j}|≤r} β^{j} > 0, we have

E[D_{ω}^{j,1}(t)D_{ω}^{j,1}(R)] ≥ c_{2} ∫_{0}^{1} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{0}^{1} dv_{1} ∫_{0}^{v_{1}} dv_{2}χ(|tu_{2} − Rv_{2}| ≤ r)
= c_{2} ∫_{0}^{1} du(1 − u) ∫_{0}^{1} dv(1 − v)χ(|tu − Rv| ≤ r)

for R ≤ t ≤ R + 1. Then, by a simple calculation, we have

inf_{R>1} inf_{R≤t≤R+1} RE[D_{ω}^{j,1}(t)D_{ω}^{j,1}(R)] > 0.

On the other hand, by (6.2), we have

RE[D_{ω}^{j,1}(R)^{2}] = \frac{1}{2} ∫_{0}^{1} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{0}^{1} dv_{1} ∫_{0}^{Rv_{1}} dv_{2}\{β^{j}(Ru_{2} − v_{2}) + β^{j}(Ru_{2} + v_{2})\}

→ \frac{1}{2} ∫_{0}^{1} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{u_{2}}^{1} dv_{1} ∫_{ℝ} β^{j}(t)dt = \frac{1}{6} ∫_{ℝ} β^{j}(t)dt

as R → ∞. Therefore we have

sup_{R>1} RE[D_{ω}^{j,1}(R)^{2}] < ∞

and

inf_{R≤t≤R+1} X^{j}(t) ≥ \frac{c_{3}}{\sqrt{R}}

for any R > 1. Using also the Schwarz inequality, we have

sup_{ηR/2≤t≤R+1} X^{j}(t) ≤ sup_{ηR/2≤t≤R+1} E[D_{ω}^{j,1}(t)^{2}]^{1/2} ≤ \frac{c_{4}}{\sqrt{ηR}}

for any R > 1. Therefore we have

p_{j}(a_{1}, a_{2}) ≥ P (\frac{\sqrt{R}}{c_{3}}(a_{1} + a_{3} + a_{4}) ≤ W_{ω}^{j} ≤ \frac{\sqrt{ηR}}{c_{4}}(a_{2} − a_{3} − a_{4}))

×P ( sup_{ηR/2≤t≤ηR or R≤t≤R+1} |\overline{D_{ω}^{j,1}}(t)| ≤ a_{4})

×P ( sup_{ηR/2≤t≤ηR or R≤t≤R+1} |D_{ω}^{j,2}(t)| ≤ a_{3})

for any a_{4} ≥ 0.

By Fernique’s theorem [26], we have

P ( sup_{ηR/2≤t≤ηR} |D_{ω}^{j,2}(t)| ≤ Ξ{ sup_{ηR/2≤t≤ηR} E[D_{ω}^{j,2}(t)^{2}]^{1/2} + 2 ∫_{1}^{∞} φ(2^{−u^{2}})du})

≤ 40 ∫_{Ξ}^{∞} e^{−u^{2}/2}du

for any Ξ ≥ \sqrt{1 + 8 log 2}, where

φ(h) = sup{E[(D_{ω}^{j,2}(t) − D_{ω}^{j,2}(s))^{2}]^{1/2} : ηR/2 ≤ t, s ≤ ηR, |t − s| ≤ ηRh/2}.

By the representation

E[D_{ω}^{j,2}(t)D_{ω}^{j,2}(s)]

= \frac{1}{2} ∫_{0}^{1} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{0}^{1} dv_{1} ∫_{0}^{v_{1}} dv_{2}\{β^{j}(tu_{2} − sv_{2}) − β^{j}(tu_{2} + sv_{2})\},

we have

E[D_{ω}^{j,2}(t)^{2}] ≤ \frac{c_{5}}{t ∨ 1}.

Since

E[(t^{2}D_{ω}^{j,2}(t) − s^{2}D_{ω}^{j,2}(s))^{2}]

= \frac{1}{2} ∫_{s}^{t} du_{1} ∫_{0}^{u_{1}} du_{2} ∫_{s}^{t} dv_{1} ∫_{0}^{v_{1}} dv_{2}(β^{j}(u_{2} − v_{2}) − β^{j}(u_{2} + v_{2}))

≤ c_{6}(t − s)^{2}t(1 ∧ t) for t > s, we have

E[(D_{ω}^{j,2}(t) − D_{ω}^{j,2}(s))^{2}] ≤ {2c_{6}|t − s|(t2 ∧ t) + 2|t^{2} − s^{2}|^{2}c_{5}/(s ∨ 1)}/t^{4}

and φ(h) ≤ c_{7}h/(1 ∨ \sqrt{ηR}). Therefore we have

P ( sup_{ηR/2≤t≤ηR} |D_{ω}^{j,2}(t)| ≥ \frac{a_{3}}{\sqrt{ηR} ∨ 1}) ≤ \frac{1}{3}

for some 0 < a_{3} < ∞. Similarly we have

P ( sup_{R≤t≤R+1} |D_{ω}^{j,2}(t)| ≥ \frac{a_{3}}{\sqrt{R} ∨ 1}) ≤ \frac{1}{3}.

Therefore we have

P ( sup_{ηR/2≤t≤ηR or R≤t≤R+1} |D_{ω}^{j,2}(t)| ≥ \frac{a_{3}}{\sqrt{ηR} ∨ 1}) ≥ \frac{1}{3}.

By the same argument, we have

P ( sup_{ηR/2≤t≤ηR or R≤t≤R+1} |\overline{D_{ω}^{j,1}}(t)| ≥ \frac{a_{4}}{\sqrt{ηR} ∨ 1}) ≥ \frac{1}{3}

for some 0 < a_{4} < ∞. With these a_{3} and a_{4}, we have

p_{j}(a_{1}, a_{2}) ≥ \frac{1}{9}P (\frac{\sqrt{R}}{c_{3}}(a_{1} + \frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1}) ≤ W_{ω}^{j} ≤ \frac{\sqrt{ηR}}{c_{4}}(a_{2} − \frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1})).

For any a_{1} > 0, we take a_{2} so that

\frac{\sqrt{ηR}}{c_{4}}(a_{2} − \frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1}) = \frac{\sqrt{R}}{c_{3}}(a_{1} + \frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1}) + 1.

Then we have

a_{2} = \frac{c_{4}}{c_{3}\sqrt{η}} a_{1} + (\frac{c_{4}}{c_{3}\sqrt{η}} + 1)\frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1} + \frac{c_{4}}{\sqrt{ηR}}  (6.3)

and

p_{j}(a_{1}, a_{2}) ≥ \frac{1}{9\sqrt{2π}} exp ( − \frac{ηR}{2c_{4}^{2}}(a_{2} − \frac{a_{3} + a_{4}}{\sqrt{ηR} ∨ 1})^{2}) ≥ c_{8} exp(−c_{9}Ra_{1}^{2} − c_{10}/η).

We now assume that the events in the probability p_{j}(a_{1}, a_{2}) with (6.3) for both j = 1 and 2 occurs. Then we have

\frac{∥A_{ω}ϕ_{n,R}^{ω}∥^{2}}{∥ϕ_{n,R}^{ω}∥^{2}} ≤ \frac{c_{11}}{ηR}(\frac{6}{η})^{2n} exp ( − R^{2}(a_{1} − \frac{c_{12}η^{3/2}}{\sqrt{ηR} ∨ 1}))

for 0 < η < c_{13} and R ≥ c_{14}. Therefore, for ∥A_{ω}ϕ_{n,R}^{ω}∥^{2}/∥ϕ_{n,R}^{ω}∥^{2} ≤ λ to hold, it is sufficient that

n ≤ \frac{R^{2}(a_{1} − c_{12}η^{3/2}/(\sqrt{ηR} ∨ 1)) + log(ληR/c_{11})}{2 log(6/η)}.

Thus, by restricting to the above events, we have

N^{−}(λ) ≥ \frac{R^{2}(a_{1} − c_{12}η^{3/2}/(\sqrt{ηR} ∨ 1)) + log(ληR/c_{11})}{2^{2}(R + 1)^{2}2 log(6/η)}

×c_{8}^{2} exp(−2c_{9}Ra_{1}^{2} − 2c_{10}/η).

By taking a_{1} = 1/\sqrt{ηR} and R = (2\sqrt{η} log(1/λ))^{2/3} so that

R^{3/2}/2 + \sqrt{η} log λ = 0,  (6.4)

we have

N^{−}(λ) ≥ c_{15} \frac{R^{3/2}(1/2 − c_{12}η^{3/2}) + \sqrt{η} log(ηR/c_{11})}{R^{2}}e^{−c_{16}/η}.  (6.5)

Since R^{3/2}(1/2 − c_{12}η^{3/2}) + \sqrt{η} log(ηR/c_{11}) ≥ R^{3/2}/3 for 0 < η < c_{17} and R ≥ c_{18}, we have

N^{−}(λ) ≥ c_{19}e^{−c_{20}/η}( log \frac{1}{λ})^{−1/3}

for 0 < λ ≤ exp(−c_{21}/\sqrt{η}).

We next consider the case that the conditions (a-i), (a-ii) and (a-vii) are satisfied. We take the vector potential as

A_{ω}^{1} = 0 and A_{ω}^{2} = \frac{d}{dx^{1}} C_{ω}^{1}(x^{1}) = ∫_{0}^{x^{1}} dtB_{ω}^{1}(t).

For any R and S > 0, we restrict the Pauli Hamiltonian to the rectangle (−R, R)×(−S, S) by the Dirichlet and the periodic boundary conditions in the first and the second components, respectively: let H_{ω}^{−,R,[S]} be the self-adjoint operator on L^{2}((−R, R) × (−S, S)) corresponding to the closure of the quadratic form (A_{ω}ϕ, A_{ω}ψ) with the domain

{ϕ ∈ C^{1}([−R, R] × [−S, S]) : ϕ(±R, ·) = 0, ϕ(·, S) = ϕ(·, −S)}.

Now the functions ϕ_{n,S}^{ω}(x) = exp(−C_{ω}^{1}(x^{1}) − 2πnz/S) satisfy the periodic condition in x^{2} and A_{ω}ϕ_{n,S}^{ω} = 0 in the sense of distributions. Then the restricted functions ϕ_{n,R,S}^{ω}(x) = ζ_{R}(x^{1})ϕ_{n,S}^{ω}(x) belong to the domain of the operator H_{ω}^{−,R+1,[S]} and satisfy

∥A_{ω}ϕ_{n,R,S}^{ω}∥^{2} ≤ c_{22}S exp ( − 2R^{2} inf_{R≤|t|≤R+1} D_{ω}^{1}(t) + 4πn(R + 1)/S)

when inf_{R≤|t|≤R+1} D_{ω}^{1}(t) ≥ 0, and

∥ϕ_{n,R,S}^{ω}∥^{2} ≥ 2SηR exp ( − 2η^{2}R^{2}( sup_{ηR/2≤|t|≤ηR} D_{ω}^{1}(t))_{+} − 4πnηR/S).

When the event in the probability p_{1}(a_{1}, a_{2}) with (6.3) occurs, we have

\frac{∥A_{ω}ϕ_{n,R,S}^{ω}∥^{2}}{∥ϕ_{n,R,S}^{ω}∥^{2}} ≤ \frac{c_{22}}{ηR} exp ( − R^{2}(a_{1} − \frac{c_{23}η^{3/2}}{\sqrt{ηR} ∨ 1}) + \frac{4π(R + 1)n}{S})

for 0 < η < c_{24}. Then, as in (6.5), the expectation of the counting function N_{ω}^{−}(λ : R, [S]) of the eigenvalues of the operator H_{ω}^{−,R,[S]} is estimated as

E[N_{ω}^{−}(λ : R + 1, [S])] ≥ c_{25} \frac{S}{R}\{R^{3/2}(1 − c_{23}η^{3/2}) + \sqrt{η} log(ηR/c_{22})\}e^{−c_{26}/η},

where R is taken as in (6.4). By the min-max principle, we see that

E[N_{ω}^{−}(λ : N(R + 1), [S])] ≥ NE[N_{ω}^{−}(λ : R + 1, [S])]

for any N ∈ ℕ. Therefore, by taking the limit as N → ∞ and using the uniqueness of the integrated density of states [19], [32], [45], we have

N^{−}(λ) ≥ \frac{c_{27}}{R^{2}} \{R^{3/2}(1 − c_{23}η^{3/2}) + \sqrt{η} log(ηR/c_{22})\}e^{−c_{26}/η}

as in the preceding case.

Acknowledgements. The author thanks Professor S. Ushiki and Professor M. Fujii for fruitful discussions on the osculatory packing. The author thanks also the referee for valuable comments.

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan e-mail: ueki@math.h.kyoto-u.ac.jp

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